Abstract Fractional Monotone Approximation, Theory and Applications (Studies in Systems, Decision and Control, 411) [1st ed. 2022] 3030959422, 9783030959425

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Studies in Systems, Decision and Control 411

George A. Anastassiou

Abstract Fractional Monotone Approximation, Theory and Applications

Studies in Systems, Decision and Control Volume 411

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

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George A. Anastassiou

Abstract Fractional Monotone Approximation, Theory and Applications

George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN, USA

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-95942-5 ISBN 978-3-030-95943-2 (eBook) https://doi.org/10.1007/978-3-030-95943-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to: “Hard working unknown mathematician”

Preface

Fractional calculus and its long history started in 1695 by L’Hospital and Leibnitz. Over the years, many different kinds of it emerged according to the needs of science given by many famous mathematicians. The last 50 years fractional calculus due to its wide applications to many applied sciences has become very popular. Its predominant kind is the newer one of Caputo type with all of its variations because of their easier applicability. These involve usually singular kernels but recently also non-singular kernels. Almost all fractional calculi engage memory in their kernels and the two divisions of singular and non-singular kernel have their own advantages and disadvantages, but both have their strong merits. In this short monograph, we employ an abstract kernel fractional calculus with applications to Prabhakar and non-singular kernel fractional calculi. Our results are univariate and bivariate. In the univariate case, we present abstract fractional monotone approximation by polynomials and splines, and in the bivariate case, we give the abstract fractional monotone constrained approximation by bivariate pseudo-polynomials and polynomials. So in this monograph, all presented is original work by the author given at a very abstract level to cover a maximum number of different fractional calculi applications. As a result, this short monograph is the natural and expected evolution of recent author’s research work put in a book form for the first time. The presented approaches are original, and chapters are self-contained and can be read independently. This monograph is suitable to be used in related graduate classes and research projects. The motivation to write this monograph came by the following: Various issues related to the modeling and analysis of ordinary and fractional-order systems have gained an increased popularity, as witnessed by many books and volumes in Springer’s program: http://www.springer.com/gp/search?query=fractional&submit= Prze%C5%9Blij, and the purpose of our book is to capture at a very general level a deeper formal analysis on some issues that are relevant to many areas for instance: decision making, complex processes, systems modeling and control, and related areas.

vii

viii

Preface

The above are deeply embedded in the fields of mathematics, engineering, computer science, physics, economics, social, and life sciences. Next listed are the author’s recent monographs in fractional analysis and applications: • George Anastassiou, “Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations”, Springer, Studies in Computational Intelligence 734, Heidelberg, New York, 2018. • George Anastassiou, “Nonlinearity: Ordinary and Fractional Approximations by Sublinear and Max-product Operators”, Springer, Studies in Systems, Decision and Control 147, Heidelberg, New York, 2018. • George Anastassiou, “Ordinary and Fractional approximation by non-additive integrals: Choquet, Shilkret and Sugeno integral approximators”, Springer, Studies in Systems, Decision and Control 190, Heidelberg, New York, 2019. • George Anastassiou, “Intelligent Analysis: Fractional Inequalities and Approximations Expanded”, Springer, Studies in Computational Intelligence 886, Heidelberg, New York, 2020, • George Anastassiou, “Generalized Fractional Calculus: New Advancements and Applications”, Springer, Studies in Systems, Decision and Control 305, Heidelberg, New York, 2021. • George Anastassiou, “Constructive Fractional Analysis with Applications”, Springer, Studies in Systems, Decision and Control 362, Heidelberg, New York, 2021. • George Anastassiou, “Unification of Fractional Calculi with Applications”, Springer, Studies in Systems, Decision and Control 398, Heidelberg, New York, 2022. The complete list of presented topics is as follows: • basic abstract fractional monotone polynomial approximation with applications, • univariate simultaneous high-order abstract fractional monotone polynomial approximation with applications, • simultaneous high-order abstract fractional monotone spline approximation and applications, • abstract bivariate left fractional pseudo-polynomial monotone constrained approximation with applications, • abstract bivariate right fractional pseudo-polynomial monotone constrained approximation and applications, and • bivariate abstract fractional monotone constrained approximation by polynomials. The book’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional approximation theory and fractional differential equations. Other possible applications can be in applied sciences like geophysics, physics, chemistry, economics, and engineering, etc. All in all what is presented here is a valuable tool for a large range of applications. Therefore, this short monograph is suitable for researchers, graduate students, practitioners, and seminars of the above disciplines, also to be in all science and engineering libraries.

Preface

ix

The preparation of the book took place during 2021 at the University of Memphis, at stay home of the author during the COVID-19 outbreak keeping him alive, happy, and in control! The author likes to thank Prof. Alina Lupas of the University of Oradea, Romania, for checking and reading the manuscript. Memphis, TN, USA December 2021

George A. Anastassiou

Contents

1 Basic Abstract Fractional Monotone Approximation . . . . . . . . . . . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Fractional Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Abstract Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 About Prabhakar Fractional Calculus . . . . . . . . . . . . . . . . . . . 1.2.3 From Generalized Non-singular Fractional Calculus . . . . . . 1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 3 4 7 11 22

2 Advanced Abstract Fractional Monotone Approximation . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fractional Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Abstract Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 About Prabhakar Fractional Calculus . . . . . . . . . . . . . . . . . . . 2.2.3 From Generalized Non-singular Fractional Calculus . . . . . . 2.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 26 26 27 29 33 45

3 Spline Abstract Fractional Monotone Approximation . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fractional Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Abstract Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 About Prabhakar Fractional Calculus . . . . . . . . . . . . . . . . . . . 3.2.3 From Generalized Non-singular Fractional Calculus . . . . . . 3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 51 51 52 53 58 71

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation by Pseudo-polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bivariate Fractional Calculi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Bivariate Abstract Fractional Calculus . . . . . . . . . . . . . . . . . .

73 73 80 80

xi

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Contents

4.2.2 About Bivariate Fractional Calculus . . . . . . . . . . . . . . . . . . . . 4.2.3 About Generalized Non-singular Fractional Calculus . . . . . . 4.2.4 Bivariate Parametrized Caputo-Fabrizio Type Non-singular Kernel Left Partial Fractional Derivative of Orders (µ1 , µ2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Right Side Abstract Bivariate Monotone Constrained Approximation by Pseudo-polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Bivariate Right Side Fractional Calculi . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Bivariate Right Abstract Fractional Calculus . . . . . . . . . . . . . 5.2.2 About Right Bivariate Fractional Calculus . . . . . . . . . . . . . . . 5.2.3 About Right Generalized Non-singular Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Bivariate Right Parametrized Caputo-Fabrizio Type Non-singular Kernel Left Partial Fractional Derivative of Orders (µ1 , µ2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Bivariate Polynomial Abstract Left and Right Fractional Monotone Constrained Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 About Bivariate Abstract Fractional Calculus . . . . . . . . . . . . . . . . . . . 6.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82 82

83 84 95 97 97 104 104 105 106

107 108 119 121 121 127 130 144

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 1

Basic Abstract Fractional Monotone Approximation

Here we extend our earlier fractional monotone approximation theory to abstract fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f ∈ C p ([−1, 1]), p ≥ 0 and let L be a linear abstract left or right fractional differential operator such that L ( f ) ≥ 0 over [0, 1] or [−1, 0], respectively. We can find a sequence of polynomials Q n of degree ≤ n such that L (Q n ) ≥ 0 over [0, 1] or [−1, 0], respectively. Additionally f is approximated quantitatively with rates uniformly by Q n with the use of first modulus of continuity of f ( p) . It follows [5].

1.1 Introduction The topic of monotone approximation initiated in 1965, [15] by O. Shisha, and became a major trend in approximation theory. The original problem was: given a positive integer k, approximate with rates a given function f whose kth derivative is ≥ 0 by polynomials (Q n )n∈N having the same property. In 1985, [6], the author and O. Shisha, continued this study by replacing the kth derivative with a linear differential operator of order k involving ordinary derivatives, again the approximation was with rates. Later in 1991, [1], the author extended this kind of study in two dimensions, etc. In 2015, [1], see Chaps. 1–8, the author went a step further, by starting the fractional monotone approximation. In that the linear differential operator is a fractional one, involving left or right side Caputo fractional derivatives. To give a flavor of it we need: Definition 1.1 ([8], p. 50) Let α > 0 and α = m ∈ N (· is the ceiling of the number). Consider f ∈ C m ([−1, 1]). We define the left side Caputo fractional derivative © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_1

1

2

1 Basic Abstract Fractional Monotone Approximation

of f of order α as follows: 

α D∗−1



1 f (x) =  (m − α)

x

(x − t)m−α−1 f (m) (t) dt,

(1.1)

−1

for any x ∈ [−1, 1], where  is the gamma function. We set 0 m f (x) = f (x) , D∗−1 f (x) = f (m) (x) , ∀x ∈ [−1, 1] . D∗−1

Also to motivate our work we mention: Theorem 1.2 ([2], p. 2) Let h, k, p be integers, 0 ≤ h ≤ k ≤ p and let f be a real  function, with f ( p) continuous in [−1, 1] and first modulus of continuity ω1 f ( p) , δ , where δ > 0. Let α j (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume for x ∈ [0, 1] that αh (x) is either ≥ some number α > 0 or ≤ some number β < 0. Let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < αj f stands for the left Caputo fractional derivative of f of order α p ≤ p. Here D∗−1 α j anchored at −1. Consider the linear left fractional differential operator L :=

k 

 αj  α j (x) D∗−1

(1.2)

j=h

and suppose, throughout [0, 1] ,

L ( f ) ≥ 0.

(1.3)

Then, for any n ∈ N, there exists a real polynomial Q n (x) of degree ≤ n such that L (Q n ) ≥ 0 throughout [0, 1] , 

and max | f (x) − Q n (x)| ≤ Cn k− p ω1

−1≤x≤1

f ( p) ,

(1.4) 1 , n

(1.5)

where C is a constant independent of n or f. As you see the monotonicity property here is true only on the critical interval [0, 1] . We will use the following important result: Theorem 1.3 (see: [17] by S.A. Teljakovskii and [18] by R.M. Trigub) Let n ∈ N. Be given a real function g, with g ( p) continuous in [−1, 1], there exists a real polynomial qn (x) of degree ≤ n such that 

( j)

j− p ( j) ( p) 1



, max g (x) − qn (x) ≤ R p n ω1 g , −1≤x≤1 n

(1.6)

1.1 Introduction

3

j = 0, 1, ..., p, where R p is a constant independent of n or g. In this chapter we perform abstract fractional calculus, left and right monotone approximation theory of Caputo type, and then we apply our results to Prabhakar fractional Calculus, generalized non-singular fractional calculus, and parametrized Caputo-Fabrizio non-singular fractional calculus. Next, we build the related necessary fractional calculi background.

1.2 Fractional Calculi Here we need to be very specific in preparation for our main results.

1.2.1 Abstract Fractional Calculus Let h, k ∈ Z+ , p ∈ N : 0 ≤ h ≤ k ≤ p. Let also N α j > 0, j = 1, ..., p, such that α0 = 0 < α1 < 1 < α2 < 2 < α3 < 3 < ... < ... < α p < p. That is α j = j, j = 1, ..., p; α0  = 0. Consider the integrable functions k j := K α j : [0, 2] → R+ , j = 0, 1, ..., p. Here g ∈ C p ([−1, 1]) . We consider the following abstract left side Caputo type fractional derivatives: k j

 αj D∗−1 g (x)

x :=

k j (x − t) g ( j) (t) dt,

(1.7)

−1

j = 1, .., p; ∀ x ∈ [−1, 1] . Similarly, we define the corresponding right side generalized Caputo type fractional derivatives: k j

 α D1−j g (x)

1 := (−1)

j

k j (t − x) g ( j) (t) dt,

(1.8)

x

j = 1, .., p; ∀ x ∈ [−1, 1] . We set

 

j j kj D∗−1 g (x) := g ( j) (x) ; k j D1− g (x) := (−1) j g ( j) (x) ,

(1.9)

for j = 1, ..., p, and also we set k0

   0 0 D∗−1 g (x) := k0 D1− g (x) := g (x) ,

(1.10)

4

1 Basic Abstract Fractional Monotone Approximation

∀ x ∈ [−1, 1] . We will assume that 1 kh (z) dz ≥ 1, when h = 0.

(1.11)

0

In the usual Caputo fractional derivatives case it is k j (z) =

z j−α j −1  , j = 1, ..., p; ∀z ∈ [0, 2] ,   j − αj

(1.12)

and (1.11) is fulfilled, by the fact that  (h − αh + 1) ≤ 1, see [2], p. 6.

1.2.2 About Prabhakar Fractional Calculus Here we follow [3, 13]. We consider the Prabhakar function (also known as the three parameter MittagLeffler function), (see [10], p. 97; [9]) γ

E α,β (z) =

∞  k=0

(γ )k zk , k! (αk + β)

(1.13)

where  is the gamma function; α, β > 0, γ ∈ R, z ∈ R, and 1 0 . (z) = (β) (γ )k = γ (γ + 1) ... (γ + k − 1). It is E α,β Let a, b ∈ R, a < b and x ∈ [a, b]; f ∈ C ([a, b]) . The left and right Prabhakar fractional integrals are defined ([3, 13]) as follows:  γ  eρ,μ,ω,a+ f (x) =

x

  γ ω (x − t)ρ f (t) dt, (x − t)μ−1 E ρ,μ

(1.14)

b

   γ γ eρ,μ,ω,b− f (x) = (t − x)μ−1 E ρ,μ ω (t − x)ρ f (t) dt,

(1.15)

a

and

x

where ρ, μ > 0; γ , ω ∈ R. Functions (1.14) and (1.15) are continuous, see [3]. Next, let f ∈ C N ([a, b]), where N = μ, (· is the ceiling of the number), 0 0. By assumption we take ρ > 0, γ < 0 and for convenience we consider only ω > 0. Therefore, we derive the basic Hardy type inequalities:     −γ eρ,N −μ,ω,−1+ 1

∞,[−1,1]

−γ

≤ 2 N −μ E ρ,N −μ+1 (2ρ ω) ,

(1.29)

1.2 Fractional Calculi

7

    −γ eρ,N −μ,ω,1− 1

and

∞,[−1,1]

−γ

≤ 2 N −μ E ρ,N −μ+1 (2ρ ω) .

(1.30)

1.2.3 From Generalized Non-singular Fractional Calculus ([4]) We need Definition 1.4 Here we use the multivariate analogue  generalized Mittag-Leffler  of function, see [14], defined for λ, γ j , ρ j , z j ∈ C, Re ρ j > 0 ( j = 1, ..., m) in terms of a multiple series of the form: E

(γ j ) (γ ,...,γ ) , ..., z m ) = E (ρ11 ,...,ρmm ),λ (z 1 , ..., z m ) = (ρ j ),λ (z 1 ∞ 

(γ1 )k1 ... (γm )km z 1k1 ...z mkm   , m k1 !...km !  k1 ,...,km =0  λ+ kjρj

(1.31)

j=1

  where γ j k j is the Pochhammer symbol,  is the gamma function. By [16], p. 157,   (1.31) converges for Re ρ j > 0, j = 1, ..., m. (γ ,...,γ )

1 m In what follows we will use the particular case of E (ρ,...,ρ),λ [ω1 t ρ , ..., ωm t ρ ], (γ j ) denoted by E (ρ),λ [ω1 t ρ , ..., ωm t ρ ], where 0 < ρ < 1, t ≥ 0, λ > 0, γ j ∈ R with       γ j k j := γ j γ j + 1 ... γ j + k j − 1 , ω j ∈ R − {0}, for j = 1, ..., m.

Let now f ∈ C n+1 ([a, b]), n ∈ Z+ . We define the Caputo type generalized left fractional derivative with non-singular kernel of order n + ρ, as n+ρ n+ρ,λ f (x) := C(γAj )(ω j ) Da∗ f (x) := Da∗

A (ρ) 1−ρ

x a

  −ωm ρ (γ j ) −ω1 ρ E (ρ),λ (x − t)ρ , ..., (x − t)ρ f (n+1) (t) dt, 1−ρ 1−ρ

(1.32)

∀ x ∈ [a, b] . Similarly, we define the Caputo type generalized right fractional derivative with non-singular kernel of order n + ρ, as

8

1 Basic Abstract Fractional Monotone Approximation n+ρ

n+ρ,λ

Db− f (x) := C(γAj )(ω j ) Db− (−1)

n+1

A (ρ) 1−ρ

b x

f (x) :=

  −ωm ρ (γ j ) −ω1 ρ E (ρ),λ (t − x)ρ , ..., (t − x)ρ f (n+1) (t) dt, 1−ρ 1−ρ

(1.33) ∀ x ∈ [a, b] . Above A (ρ) is a normalizing constant. The above derivatives (1.32), (1.33) generalize the Atangana-Baleanu fractional derivatives [7]. We mention the following Hardy type inequalities: Theorem 1.5 ([4]) All as above with γ j > 0, j = 1, ..., m; λ = 1. Then  n+ρ   n+ρ   (b − a) |A (ρ)| D    a∗ f ∞ , Db− f ∞ ≤ 1−ρ     |ωm | ρ (γ j ) |ω1 | ρ ρ ρ  (n+1)  f < ∞, E (ρ),2 (b − a) , ..., (b − a) ∞ 1−ρ 1−ρ

(1.34)

where n ∈ Z+ . We also mention Theorem 1.6 ([4]) All as above with γ j > 0, j = 1, ..., m, and λ > 0, 0 < ρ < 1, etc. Then n+ρ n+ρ f, Db− f ∈ C ([a, b]) , n ∈ Z+ . Da∗ We rewrite (1.32) and (1.33), and for [a, b] = [−1, 1] . Let μ > 0 with μ ∈ / N and μ = n ∈ N. That is 0 < 1 − n + μ < 1, and let f ∈ C n ([−1, 1]). Then, we have μ

μ,λ

D−1∗ f (x) := C(γAj )(ω j ) D−1∗ f (x) := A (1 − n + μ) n−μ

x −1

 −ω1 (1 − n + μ) (γ j ) E (1−n+μ),λ (x − t)1−n+μ , ..., n−μ

 −ωm (1 − n + μ) (x − t)1−n+μ f (n) (t) dt, n−μ and

μ

μ,λ

D1− f (x) := C(γAj )(ω j ) D1− f (x) :=

(1.35)

1.2 Fractional Calculi

9

A (1 − n + μ) (−1) n−μ

1

n

x

 −ω1 (1 − n + μ) (γ j ) E (1−n+μ),λ (t − x)1−n+μ , ..., (1.36) n−μ

 −ωm (1 − n + μ) 1−n+μ f (n) (t) dt, (t − x) n−μ ∀ x ∈ [−1, 1] . m m 0 0 f = f , D1− f = f , and D−1∗ f = f (m) , D1− f = (−1)m f (m) , We will set D−1∗ when m ∈ N. We make Remark 1.7 Fractional Calculi of Sects. 1.2.2 and 1.2.3 are special cases of abstract fractional calculus, see Sect. 1.2.1. In particular the important condition (1.11) is fulfilled. 1 So, we will verify 0 kh (z) dz ≥ 1, h = 0. (I) First for Sect. 1.2.2: We notice that 1 −γ z N −μ−1 E ρ,N −μ (ωz ρ ) dz = 0

(here ρ, N − μ > 0, γ < 0, ω > 0) 1

∞ 

z N −μ−1

k=0

0 ∞  k=0

∞  k=0 ∞  k=0

(−γ )k (ωz ρ )k dz k! (ρk + N − μ)

(−γ )k ωk k! (ρk + N − μ)

(−γ )k ωk k! (ρk + N − μ)

1

(by [11], p. 175)

=

z N −μ−1 z ρk dz =

(1.37)

0

1

z (ρk+N )−μ−1 dz =

0

(−γ )k ωk −γ = E ρ,N −μ+1 (ω) ≥ 1, k! (ρk + N − μ + 1)

for suitable ω > 0. (II) Next, for Sect. 1.2.3: Here γ j > 0, j=1, ..., m; λ = 1; N μ > 0, μ = n ∈ N, ω j < 0, j = 1, ..., m. Without loss of generality we assume that A (1 − n + μ) > 0.

10

1 Basic Abstract Fractional Monotone Approximation

We have that A (1 − n + μ) n−μ

1



(γ j )

E (1−n+μ),1 0

−ω1 (1 − n + μ) 1−n+μ z , n−μ

 −ωm (1 − n + μ) 1−n+μ dz = z ..., n−μ (here 0 < 1 − (n − μ) = 1 − n + μ < 1) ⎡ A (1 − n + μ) n−μ

1 ⎢  ⎢ ∞ ⎢ ⎢ ⎣k1 ,...,km =0 0

(γ1 ) ... (γm )km  k1   m   1+ k j (1 − n + μ) 

j=1



m j=1

−ω j (1−n+μ) n−μ

k j

z

(1−n+μ)

m j=1

kj

⎤

k1 !...km !  = ∞  k1 ,...,km =0

 =

=

A (1 − n + μ) n−μ

(γ1 ) ... (γm ) 

 k 1  k m m  1+ j=1 k j (1 − n + μ)

A (1 − n + μ) n−μ



∞  k1 ,...,km =0

m j=1



 −ω j (1−n+μ) k j n−μ

k1 !...km !

(γ1 ) ... (γm )

 k 1  k m  m  2+ j=1 k j (1 − n + μ)

A (1 − n + μ) (γ j ) E (1−n+μ),2 n−μ



⎥ ⎥ dz ⎦

1

z (1−n+μ)

m j=1

kj

dz

0

m j=1

(1.38)

 −ω j (1−n+μ) k j n−μ

k1 !...km !

−ω1 (1 − n + μ) −ωm (1 − n + μ) , ..., n−μ n−μ

≥ 1, (1.39)

for suitable ω j < 0, for j = 1, ..., m. We also need Definition 1.8 Let f ∈ C n ([−1, 1]), N μ > 0, μ = n ∈ N; ω < 0. That is 0 < 1 − n + μ < 1. The parametrized Caputo-Fabrizio non-singular kernel fractional derivatives, left and right of order μ, respectively, are given as follows (see also [12]):

1.2 Fractional Calculi CF μ ω D−1+

CF μ ω D1−

11

1 f (x) := n−μ

f (x) :=

(−1)n n−μ

x −1

1 x

 (1 − n + μ) ω exp − (x − t) f (n) (t) dt, n−μ

(1.40)

 (1 − n + μ) ω exp − (t − x) f (n) (t) dt, n−μ

(1.41)

∀ x ∈ [−1, 1] . Equations (1.40), (1.41) are special cases of (1.7), (1.8). We make Remark 1.9 We want to evaluate 1 ∞> 0

 (1 − n + μ) ω z dz exp − n−μ

(call δ := − (1−n+μ)ω ) n−μ 1 = 0

∞ 

 eδ 1 1 1 δ e −1 = − = eδz dz = eδz |10 = δ δ δ δ

=

k=0

δ

δk k!

−

1 δ

(1.42)

⎞ ⎛ k−1 1−n+μ k−1 ∞ (−ω)  ⎟ ⎜ n−μ = ⎠ ≥ 1, ⎝ k! k! k=0

∞  δ k−1 k=0

for suitable ω < 0. So, again condition (1.11) is fulfilled.

1.3 Main Results We give Theorem 1.10 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p ∈ N and let f be a real  function, f ( p) is continuous in [−1, 1] with modulus of continuity ω1 f ( p) , δ , δ > 0. Let α j (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume for x ∈ [0, 1] that αh (x) is either ≥ some number α > 0 or ≤ some number β < 0. Let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < α p < p. Here we

12

1 Basic Abstract Fractional Monotone Approximation

adopt the abstract fractional calculus terminology and assumptions from above. So kj αj D∗−1 f stands for the abstract left Caputo type fractional derivative of order α j anchored at −1. We consider the linear abstract left fractional differential operator L :=

k 

α j (x)

k j

α

j D∗−1



(1.43)

j=h

and suppose, throughout [0, 1], L ( f ) ≥ 0. Then, for any n ∈ N, there exists a real polynomial Q n (x) of degree ≤ n such that L (Q n ) ≥ 0 throughout [0, 1] , 

and max | f (x) − Q n (x)| ≤ Cn

k− p

−1≤x≤1

ω1

f

(1.44)

( p)

1 , , n

(1.45)

where C is independent of n or f . Proof Let n ∈ N. By Theorem 1.3 given a real function g, with g ( p) continuous in [−1, 1], there exists a real polynomial qn (x) of degree ≤ n such that 

( j) j− p ( j) ( p) 1



, max g (x) − qn (x) ≤ R p n ω1 g , −1≤x≤1 n

(1.46)

j = 0, 1, ..., p, where R p is independent of n or g. We notice that (x ∈ [−1, 1])

k α j   

j D g (x) − k j D α j qn (x) = ∗−1 ∗−1

x



x



k j (x − t) g ( j) (t) dt − k j (x − t) q ( j) (t) dt = n





−1

−1

x





 ( j) 

k j (x − t) g (t) − q ( j) (t) dt ≤ n





−1

x −1



(1.46) k j (x − t) g ( j) (t) − qn( j) (t) dt ≤

(1.47)

1.3 Main Results

13

⎛ ⎝

x

−1

⎞

 j− p ( p) 1 ⎠ k j (x − t) dt R p n ω1 g , = n

⎛ 2 ⎛ x+1 ⎞ ⎞     1 j− p ( p) 1 ⎝ ⎝ k j (z) dz ⎠ R p n j− p ω1 g ( p) , ⎠ ≤ . k j (z) dz R p n ω1 g , n n 0

0

We have proved that

k α j  



j D g (x) − k j D α j qn (x) ≤ ∗−1 ∗−1

(1.48)

⎛ 2 ⎞   ⎝ k j (z) dz ⎠ R p n j− p ω1 g ( p) , 1 , ∀ x ∈ [−1, 1] . n 0

That is

   

αj αj g (x) − k j D∗−1 qn (x) ≤ max k j D∗−1

−1≤x≤1

⎛ ⎝

2

⎞

 1 k j (z) dz ⎠ R p n j− p ω1 g ( p) , , j = 1, ..., p. n

0

So we have 

 

  1 αj αj , g (x) − k j D∗−1 qn (x) ≤ λ j R p n j− p ω1 g ( p) , max k j D∗−1 −1≤x≤1 n where

(1.49)

2 λ j :=

k j (z) dz, j = 1, ..., p.

(1.50)

0

Inequality (1.49) is valid when j = 0 by (1.46), and we can set λ0 = 1. Put



s j ≡ sup αh−1 (x) α j (x) , j = h, ..., k, −1≤x≤1

and

 ηn := R p ω1

⎞ ⎛  k 1 ⎝ f ( p) , s j λ j n j− p ⎠ . n j=h

(1.51)

(1.52)

I. Suppose, throughout [0, 1], αh (x) ≥ α > 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n so that

14

1 Basic Abstract Fractional Monotone Approximation

   (1.49) αj  αj f (x) + ηn (h!)−1 x h − k j D∗−1 max k j D∗−1 Q n (x) ≤

−1≤x≤1



f ( p) ,

λ j R p n j− p ω1

1 , n

(1.53)

j = 0, 1, ..., p. In particular ( j = 0) holds



(1.53)  max f (x) + ηn (h!)−1 x h − Q n (x) ≤ R p n − p ω1

 f

−1≤x≤1



and −1

max | f (x) − Q n (x)| ≤ ηn (h!)

−1≤x≤1

−1

(h!)

 R p ω1

+ Rpn

−p

ω1

f

( p)

( p)

1 , , n

1 , n

=

⎞ ⎛   k 1 j− p ⎠ −p ( p) ( p) 1 ⎝ f , ≤ sjλjn + R p n ω1 f , n n j=h 

R p ω1

(1.54)

(1.55)

⎛ ⎞ k  1 f ( p) , n k− p ⎝1 + (h!)−1 sjλj⎠ . n j=h

That is ⎛ max | f (x) − Q n (x)| ≤ R p ⎝1 + (h!)−1

−1≤x≤1

k 

⎞ s j λ j ⎠ n k− p ω1

j=h

proving (1.45). Here L=

k 

α j (x)

k j

 f

( p)

1 , (1.56) , n

αj  , D∗−1

j=h

and suppose, throughout [0, 1], L f ≥ 0. So over 0 ≤ x ≤ 1, using (1.52) and (1.53), we have ηn k h α h h αh−1 (x) L (Q n (x)) = αh−1 (x) L ( f (x)) + D∗−1 x + h! k  j=h

αh−1 (x) α j (x)

%

kj

α

j D∗−1 Q n (x) −

kj

α

j D∗−1 f (x) −

ηn h!

kj

& αj D∗−1 xh ≥

(1.57)

1.3 Main Results

15

⎛ ηn h!

kh

αh D∗−1 xh

−⎝

k 

⎞ sjλjn

j− p ⎠

 R p ω1

f

( p)

j=h

1 , n

=

ηn h!

kh

αh D∗−1 x h − ηn =: ϕ

(if h = 0, then αh = 0, and ϕ = 0). If h = 0, then ϕ = ηn

 kh

αh D∗−1 xh

h!

⎛ x ⎞  − 1 = ηn ⎝ kh (x − t) dt − 1⎠ = −1

⎛ x+1 ⎛ 1 ⎞ ⎞   ηn ⎝ kh (z) dz − 1⎠ ≥ ηn ⎝ kh (z) dz − 1⎠ ≥ 0 0

(1.58)

0

by the assumption (1.11):

1

kh (z) dz ≥ 1, when h = 0.

0

Hence in both cases we get L (Q n (x)) ≥ 0, x ∈ [0, 1] .

(1.59)

II. Suppose, throughout [0, 1], αh (x) ≤ β < 0. In this case let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n such that

   (1.49) αj  αj f (x) − ηn (h!)−1 x h − k j D∗−1 Q n (x) ≤ max k j D∗−1

−1≤x≤1

 λ j R p n j− p ω1

f ( p) ,

1 , n

(1.60)

j = 0, 1, ..., p. In particular ( j = 0) holds

(1.60)

  max f (x) − ηn (h!)−1 x h − Q n (x) ≤ R p n − p ω1

 f

−1≤x≤1

( p)

1 , , n

(1.61)

and −1

max | f (x) − Q n (x)| ≤ ηn (h!)

−1≤x≤1

 R p ω1

f ( p) ,



+ Rpn

−p

 ω1

⎛

1 n k− p ⎝1 + (h!)−1 n

f

( p)

k  j=h

1 , n



⎞ sjλj⎠ .

(as before)

≤

(1.62)

16

1 Basic Abstract Fractional Monotone Approximation

That is (1.45) is again true. Again suppose, throughout [0, 1], L f ≥ 0. Also if 0 ≤ x ≤ 1, then αh−1 (x) L (Q n (x)) = αh−1 (x) L ( f (x)) − k 

αh−1 (x) α j (x)

%

kj

α

j D∗−1 Q n (x) −

kj

ηn h!

α

j D∗−1 f (x) +

j=h

⎛ −

ηn h!

kh

αh D∗−1 xh + ⎝

k 

⎞ s j λ j n j− p ⎠ R p ω1



f ( p) ,

j=h

1 n

=−

kh

αh D∗−1 xh+

& (1.60) ≤

ηn h!

kj

j D∗−1 xh

ηn h!

kh

αh D∗−1 x h + ηn =: ψ

α

(if h = 0, then αh = 0, and ψ = 0). If h = 0, then  ψ = ηn 1 −

kh

αh D∗−1 xh h!



⎡ = ηn ⎣1 −

x

⎤ kh (x − t) dt ⎦ =

(1.63)

−1

⎡

⎡ ⎤ ⎤ x+1 1 ηn ⎣1 − kh (z) dz ⎦ ≤ ηn ⎣1 − kh (z) dz ⎦ ≤ 0. 0

0

Hence again in both cases L (Q n (x)) ≥ 0, ∀ x ∈ [0, 1] .

(1.64) 

We also present Theorem 1.11 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p ∈ N, where h is even, and let f be a real function, f ( p) is continuous in [−1, 1] with modulus of continuity ω1 f ( p) , δ , δ > 0. Let α j (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume for x ∈ [−1, 0] that αh (x) is either ≥ some number α > 0 or ≤ some number β < 0. Let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < α p < p. Here we adopt the abstract fractional calculus terminology and α assumptions from above. So k j D1−j f stands for the abstract right Caputo type fractional derivative of order α j anchored at 1. We consider the linear abstract right fractional differential operator k   α  (1.65) L := α j (x) k j D1−j j=h

and suppose, throughout [−1, 0],

1.3 Main Results

17

L ( f ) ≥ 0. Then, for any n ∈ N, there exists a real polynomial Q n (x) of degree ≤ n such that L (Q n ) ≥ 0 throughout [−1, 0] , 

and max | f (x) − Q n (x)| ≤ Cn

−1≤x≤1

k− p

ω1

f

( p)

(1.66) 1 , , n

(1.67)

where C is independent of n or f . Proof Let x ∈ [−1, 1], we observe that

k α j  



j D g (x) − k j D α j qn (x) = 1− 1−

1



1



k j (t − x) g ( j) (t) dt − k j (t − x) q ( j) (t) dt = n





x

x

1





 ( j) 

k j (t − x) g (t) − q ( j) (t) dt ≤ n





x

1



(1.6) k j (t − x) g ( j) (t) − qn( j) (t) dt ≤

(1.68)

x

⎛ ⎝

1

⎞

 j− p ( p) 1 ⎠ = k j (t − x) dt R p n ω1 g , n

x

⎛ 2 ⎛ 1−x ⎞ ⎞     1 1 ⎝ k j (z) dz ⎠ R p n j− p ω1 g ( p) , ≤ ⎝ k j (z) dz ⎠ R p n j− p ω1 g ( p) , . n n 0

0

That is we have derived

 

  α α max k j D1−j g (x) − k j D1−j qn (x) ≤

−1≤x≤1

⎛ ⎝

2 0

⎞

 j− p ( p) 1 ⎠ , j = 1, ..., p. k j (z) dz R p n ω1 g , n

(1.69)

18

1 Basic Abstract Fractional Monotone Approximation

We call

2 λ j :=

k j (z) dz, j = 1, ..., p.

(1.70)

0

Therefore we can write 

k α j  k j α j 

j− p ( p) 1 j



, D1− g (x) − D1− qn (x) ≤ λ j R p n ω1 g , max −1≤x≤1 n for j = 1, ..., p. Inequality (1.71) is valid when j = 0 by (1.6), so we can set λ0 = 1. Put



s j ≡ sup αh−1 (x) α j (x) , j = h, ..., k, −1≤x≤1

and

 ηn := R p ω1

⎞ ⎛  k 1 ⎝ f ( p) , s j λ j n j− p ⎠ . n j=h

(1.71)

(1.72)

(1.73)

I. Suppose, throughout [−1, 0], αh (x) ≥ α > 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n so that

   (1.71) α  α max k j D1−j f (x) + ηn (h!)−1 x h − k j D1−j Q n (x) ≤

−1≤x≤1

 λ j Rpn

j− p

ω1

f

( p)

1 , , n

(1.74)

j = 0, 1, ..., p. In particular ( j = 0) holds



(1.74)  max f (x) + ηn (h!)−1 x h − Q n (x) ≤ R p n − p ω1



−1≤x≤1

f ( p) ,

1 , n

(1.75)

and, as earlier, ⎛ max | f (x) − Q n (x)| ≤ R p ⎝1 + (h!)−1

−1≤x≤1

k 

⎞ s j λ j ⎠ n k− p ω1

j=h

proving (1.67). Here L=

k  j=h

α j (x)

k j

α  D1−j ,



f ( p) ,

1 , (1.76) n

1.3 Main Results

19

and suppose, throughout [−1, 0], L f ≥ 0. So over −1 ≤ x ≤ 0, we get ηn h!

αh−1 (x) L (Q n (x)) = αh−1 (x) L ( f (x)) + k 

αh−1 (x) α j (x)

%

kj

α

D1−j Q n (x) −

kj

α

D1−j f (x) −

j=h

⎛ ηn h!

kh

αh h D1− x −⎝

k 

⎞ s j λ j n j− p ⎠ R p ω1



1 n

f ( p) ,

j=h

ηn h!

=

kh

kj

ηn h!

αh h D1− x +

α

D1−j x h

kh

& (1.74) ≥

(1.77)

αh h D1− x − ηn =: ξ

(if h = 0, then αh = 0, and ξ = 0). If h = 0, then ξ = ηn

 kh

⎛ ⎞ 1 αh h D1− x − 1 = ηn ⎝(−1)h kh (t − x) dt − 1⎠ = h! x

(h is even) ⎛ 1 ⎛ 1−x ⎞ ⎞   ηn ⎝ kh (t − x) dt − 1⎠ = ηn ⎝ kh (z) dz − 1⎠ ≥ x

0

⎛ ηn ⎝

1

⎞ kh (z) dz − 1⎠ ≥ 0,

(1.78)

0

by the assumption (1.11):

1

kh (z) dz ≥ 1, when h = 0.

0

Hence in both cases we get L (Q n (x)) ≥ 0, x ∈ [−1, 0] .

(1.79)

II. Suppose, throughout [−1, 0], αh (x) ≤ β < 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n so that

   (1.71) α  α max k j D1−j f (x) − ηn (h!)−1 x h − k j D1−j Q n (x) ≤

−1≤x≤1

 λ j Rpn

j− p

ω1

f

( p)

1 , , n

(1.80)

20

1 Basic Abstract Fractional Monotone Approximation

j = 0, 1, ..., p. In particular ( j = 0) holds 



(1.80)  max f (x) − ηn (h!)−1 x h − Q n (x) ≤ R p n − p ω1

f

−1≤x≤1

( p)

1 , , n

(1.81)

and as earlier, ⎛ max | f (x) − Q n (x)| ≤ R p ⎝1 + (h!)

−1

k 

−1≤x≤1

⎞ s j λ j ⎠ n k− p ω1



( p)

f

j=h

1 , (1.82) , n

proving (1.67). Again suppose, throughout [−1, 0], L f ≥ 0. Also if −1 ≤ x ≤ 0, then ηn h!

αh−1 (x) L (Q n (x)) = αh−1 (x) L ( f (x)) − k 

αh−1 (x) α j (x)

%

kj

α

D1−j Q n (x) −

kj

α

D1−j f (x) +

j=h

⎛ −

ηn h!

kh

αh h D1− x +⎝

k 

⎞ s j λ j n j− p ⎠ R p ω1



f ( p) ,

j=h

 = ηn 1 −

kh

αh h D1− x h!

1 n

ηn h!

kh

kj

αh h D1− x +

α

D1−j x h

=−

ηn h!

kh

& (1.80) ≤

(1.83)

αh h D1− x + ηn

=: ρ

(if h = 0, then αh = 0, and ρ = 0). If h = 0, then ⎛ ρ = ηn ⎝1 −

1

⎞

⎛

⎞ 1−x kh (t − x) dt ⎠ = ηn ⎝1 − kh (z) dz ⎠ ≤

x

(1.84)

0

⎛ ηn ⎝1 −

1

⎞ kh (z) dz ⎠ ≤ 0.

0

Hence in both cases we get again L (Q n (x)) ≥ 0, ∀ x ∈ [−1, 0] .

(1.85)

1.3 Main Results

21

Conclusion 1.12 Clearly Theorem 1.10 generalizes Theorem 1.2, and Theorem 1.11 generalizes Theorem 2.2, p. 12 of [2]. Furthermore there, the approximating polynomial Q n depends on f, ηn , h; which ηn depends on n, R p , n, k, s j , λ j ; which λ j depends on k j . I.e. polynomial Q n among others depends on the type of fractional calculus we use. Consequently, Theorem 1.10 is valid for the following left fractional linear differential operators: (1) k & %  γ (1.86) α j (x) C Dρ,α j ,ω,−1+ , L 1 := j=h

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (1.16)); (2) k   αj  L 2 := α j (x) D−1∗ , (1.87) j=h

(see (1.35)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and (3) k   αj  α j (x) Cω F D−1+ , (1.88) L 3 := j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Similarly, Theorem 1.11 is valid for the following right fractional linear differential operators: (1)∗ k & %  γ (1.89) L ∗1 := α j (x) C Dρ,α j ,ω,1− , j=h

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (1.17)); (2)∗ k   α  (1.90) L ∗2 := α j (x) D1−j , j=h

(see (1.36)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and

22

1 Basic Abstract Fractional Monotone Approximation

(3)∗ L ∗3 :=

k 

α j (x)

C F ω

α  D1−j ,

(1.91)

j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Our developed abstract fractional monotone approximation theory with its applications involves weaker conditions than the one with ordinary derivatives ([6]), and can cover many diverse general cases in a multitude of complex settings and environments.

References 1. Anastassiou, G.A.: Bivariate Monotone Approximation. Proc. Amer. Math. Soc. 112(4), 959– 964 (1991) 2. Anastassiou, G.A.: Frontiers in Approximation Theory. World Scientific Publishing Co Pte Ltd., New Jersey, Singapore (2015) 3. Anastassiou, G.A.: Foundations of Generalized Prabhakar-Hilfer fractional Calculus with Applications. Cubo (2021, accepted) 4. Anastassiou, G.A.: Multiparameter fractional differentiation with non singular kernel. Issues Anal (2021, accepted) 5. Anastassiou, G.A.: Abstract fractional monotone approximation with applications (2021, submitted) 6. Anastassiou, G.A., Shisha, O.: Monotone approximation with linear differential operators. J. Approx. Theor. 44, 391–393 (1985) 7. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) 8. Diethelm, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Vol. 2004, 1st edn. Springer, New York, Heidelberg (2010) 9. Giusti, A., et al.: A practical guide to Prabhakar fractional calculus. Fractional Calc. Appl. Anal. 23(1), 9–54 (2020) 10. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Heidelberg, New York (2014) 11. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York, Heidelberg, Berlin (1965) 12. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015) 13. Polito, F., Tomovski, Z.: Some properties of Prabhakar-type fractional calculus operators. Fractional Differ. Calc. 6(1), 73–94 (2016) 14. Saxena, R.K., Kalla, S.L., Saxena, R.: Integr. Transf. Spec. Funct. Multivariate analogue of generalized Mittag-Leffler function 22(7), 533–548 (2011) 15. Shisha, O.: Monotone approximation. Pacific J. Math. 15, 667–671 (1965) 16. Srivastava, H.M., Daoust, M.C.: A note on the convergence of Kompe’ de Feriet’s double hypergeometric series. Math. Nachr. 53, 151–159 (1972) 17. Teljakovskii, S.A.: Two theorems on the approximation of functions by algebraic polynomials. Mat. Sb. 70(112), 252–265 (1966) [Russian]; Amer. Math. Soc. Trans. 77(2), 163–178 (1968) 18. Trigub, R.M.: Approximation of functions by polynomials with integer coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 26, 261–280 (1962) [Russian]

Chapter 2

Advanced Abstract Fractional Monotone Approximation

Here we extend our earlier univariate high order simultaneous fractional monotone approximation theory to abstract univariate high order simultaneous fractional monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f ∈ C r ([−1, 1]), r ≥ 0 and let L ∗ be a linear abstract left or right fractional differential operator such that L ∗ ( f ) ≥ 0 over [0, 1] or [−1, 0], respectively. We can find a sequence of polynomials Q n of degree ≤ n such that L ∗ (Q n ) ≥ 0 over [0, 1] or [−1, 0], furthermore f is approximated left or right fractionally and simultaneously by Q n on [−1, 1]. The degree of these restricted approximations is given quantitatively by inequalities using a higher order modulus of smoothness for f (r ) . It follows [6].

2.1 Introduction The topic of monotone approximation started in 1965, [17] by O. Shisha, and became a major part of approximation theory. The original problem was: given a positive integer k, approximate with rates a given function f whose kth derivative is ≥ 0 by polynomials (Q n )n∈N having the same property. In 1985, [7], the author and O. Shisha, continued this study by replacing the kth derivative with a linear differential operator of order k involving ordinary derivatives, again the approximation was with rates. Later in 1991, [1], the author extended this kind of study in two dimensions, etc. We use also the notation I = [−1, 1]. We would like to mention

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_2

23

24

2 Advanced Abstract Fractional Monotone Approximation

Theorem 2.1 (Gonska and Hinnemann [11]). Let r ≥ 0 and s ≥ 1. Then there exists of linear polynomial operators mapping C r (I ) into Pn a sequence Q n = Q (r,s) n (space of polynomials of degree ≤ n), such that for all f ∈ C r (I ), all |x| ≤ 1 and all n ≥ max (4 (r + 1) , r + s) we have   (k)    f (x) − (Q n f )(k) (x) ≤ Mr,s (n (x))r −k ωs f (r ) , n (x) , 0 ≤ k ≤ r, (2.1) √

2

where n (x) = 1−x + n12 , and Mr,s is a constant independent of f , x, and n. Above n ωn is the usual modulus of smoothness of order s with respect to the supremum norm. Theorem 2.1 implies the useful Corollary 2.2 ([2]) Let r ≥ 0 and s ≥ 1. Then there exists a sequence Q n = Q (r,s) n of linear polynomial operators mapping C (r ) (I ) into Pn , such that for all f ∈ C r (I ) and all n ≥ max (4 (r + 1) , r + s) we have   (k)  f − (Q n f )(k)  ≤ Cr,s ωs ∞ n r −k

 f

(r )

 1 , k = 0, 1, ..., r, , n

(2.2)

where Cr,s is a constant independent of f and n. In [2] we proved the motivational Theorem 2.3 Let h, v, r be integers, h ≤ v ≤ r and let f ∈ C r (I ), with f (r )  0(r≤ ) having modulus of smoothness ωs f , δ there, s ≥ 1. Let α j (x), j = h, h + 1, ..., v be real functions, defined and bounded on I and suppose αh is either ≥ α > 0 or ≤ β < 0 throughout I . Take the operator L=

v 

α j (x)

j=h

dj dx j

(2.3)

and assume, throughout I, L ( f ) ≥ 0.

(2.4)

Then, for every integer n ≥ max (4 (r + 1) , r + s), there exists a real polynomial Q n (x) of degree ≤ n such that L (Q n ) ≥ 0 throughout I, and

  (k)  f − Q (k)  n

∞

≤

C n r −v

 ωs

f

(r )

 1 , 0 ≤ k ≤ h. , n

(2.5)

(2.6)

2.1 Introduction

25

Moreover, we set   (k)  f − Q (k)  n

∞

≤

C

n



ω r −k s

f (r ) ,

 1 , h + 1 ≤ k ≤ r, n

(2.7)

where C is a constant independent of f and n. In [3], Chap. 3, we extended Theorem 2.3 to the fractional level. Indeed there L is replaced by L ∗ , a linear left Caputo fractional differential operator. Then the monotonicity property is only true on the critical interval [0, 1]. Simultaneous and fractional convergence remained true on all of I . We need Definition 2.4 ([9], p. 50) Let α > 0 and α = m, (· ceiling of the number). Consider f ∈ C m ([−1, 1]). We define the left Caputo fractional derivative of f of order α as follows:  x   α 1 (2.8) D∗−1 f (x) = (x − t)m−α−1 f (m) (t) dt,  (m − α) −1 for any x ∈ [−1, 1], where  is the gamma function. We set 0 f (x) = f (x) , D∗−1 m f (x) = f (m) (x) , ∀ x ∈ [−1, 1] . D∗−1

(2.9)

So in [3], chap. 3, we proved the following result: Theorem 2.5 Let h, v, r be integers, 1≤ h ≤v ≤ r and let f ∈ C r ([−1, 1]), with f (r ) having modulus of smoothness ωs f (r ) , δ there, s ≥ 1. Let α j (x), j = h, h + 1, ..., v be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. Let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < αj f stands for the left Caputo fractional derivative of f of ... < αr ≤ r . Here D∗−1 order α j anchored at −1. Consider the linear left fractional differential operator ∗

L :=

v 

αj α j (x) D∗−1

(2.10)

j=h

and suppose, throughout [0, 1] ,

L ∗ ( f ) ≥ 0.

(2.11)

Then, for any n ∈ N such that n ≥ max (4 (r + 1) , r + s), there exists a real polynomial Q n (x) of degree ≤ n such that L ∗ (Q n ) ≥ 0 throughout [0, 1] ,

(2.12)

26

2 Advanced Abstract Fractional Monotone Approximation

and  α j     αj f (x) − D∗−1 Q n (x) ≤ sup  D∗−1

−1≤x≤1

2 j−α j Cr,s  ωs  j − α j + 1 nr − j 



f (r ) ,

j = h + 1, ..., r ; Cr,s is a constant independent of f and n. Set   l j :≡ sup αh−1 (x) α j (x) , h ≤ j ≤ v.

 1 , n (2.13)

(2.14)

x∈[−1,1]

When j = 1, ..., h we derive  α j     αj Cr,s sup  D∗−1 f (x) − D∗−1 Q n (x) ≤ r −v ωs n −1≤x≤1  v 

2τ −ατ lτ  (τ − ατ + 1) τ =h

  h− j  λ=0

2h−α j −λ   λ! h − α j − λ + 1



 f

(r )

 1 , . n

 2 j−α j  . +   j − αj + 1 (2.15)

Finally it holds Cr,s sup | f (x) − Q n (x)| ≤ r −v ωs n −1≤x≤1

 f

(r )

1 , n



 v 1  2τ −ατ +1 . lτ h! τ =h  (τ − ατ + 1) (2.16)

In this chapter we establish abstract fractional calculus left and right high order monotone approximation theory of Caputo type, and then we apply our results to Prabhakar fractional Calculus, generalized non-singular fractional calculus, and parametrized Caputo-Fabrizio non-singular fractional calculus. Next, we build the related necessary fractional calculi background.

2.2 Fractional Calculi Here we need to be very specific in preparation for our main results.

2.2.1 Abstract Fractional Calculus Let h, v ∈ Z+ , r ∈ N : 0 ≤ h ≤ v ≤ r . Let also N α j > 0, j = 1,..., r, such that α0 = 0 < α1 < 1 < α2 < 2 < α3 < 3 < ... < ... < αr < r . That is α j = j, j = 1, ..., r ; α0 = 0.

2.2 Fractional Calculi

27

Consider the integrable functions k j := K α j : [0, 2] → R+ , j = 0, 1, ..., r . Here g ∈ C r ([−1, 1]) . We consider the following abstract left side Caputo type fractional derivatives: k j

 αj D∗−1 g (x)

x :=

k j (x − t) g ( j) (t) dt,

(2.17)

−1

j = 1, .., r ; ∀ x ∈ [−1, 1] . Similarly, we define the corresponding right side generalized Caputo type fractional derivatives: k j

 α D1−j g (x)

1 := (−1)

j

k j (t − x) g ( j) (t) dt,

(2.18)

x

j = 1, .., r ; ∀ x ∈ [−1, 1] . We set     j j kj D∗−1 g (x) := g ( j) (x) ; k j D1− g (x) := (−1) j g ( j) (x) ,

(2.19)

for j = 1, ..., r , and also we set k 0

   0 0 D∗−1 g (x) := k0 D1− g (x) := g (x) ,

(2.20)

∀ x ∈ [−1, 1] . We will assume that 1 kh (z) dz ≥ 1, when h = 0.

(2.21)

0

In the usual Caputo fractional derivatives case it is k j (z) =

z j−α j −1  , j = 1, ..., r ; ∀ z ∈ [0, 2] ,   j − αj

and (2.21) is fulfilled, by the fact that  (h − αh + 1) ≤ 1, see [3], p. 6.

2.2.2 About Prabhakar Fractional Calculus Here we follow [4, 15].

(2.22)

28

2 Advanced Abstract Fractional Monotone Approximation

We consider the Prabhakar function (also known as the three parameter MittagLeffler function), (see [12], p. 97; [10]) γ

E α,β (z) =

∞  k=0

(γ )k zk , k! (αk + β)

(2.23)

where  is the gamma function; α, β > 0, γ ∈ R, z ∈ R, and 1 0 . (z) = (β) (γ )k = γ (γ + 1) ... (γ + k − 1). It is E α,β Let a, b ∈ R, a < b and x ∈ [a, b]; f ∈ C ([a, b]) . The left and right Prabhakar fractional integrals are defined ([4, 15]) as follows:  γ  eρ,μ,ω,a+ f (x) =

x



γ ω (x − t)ρ f (t) dt, (x − t)μ−1 E ρ,μ

(2.24)

b  

γ γ eρ,μ,ω,b− f (x) = (t − x)μ−1 E ρ,μ ω (t − x)ρ f (t) dt,

(2.25)

a

and

x

where ρ, μ > 0; γ , ω ∈ R. Functions (2.24) and (2.25) are continuous, see [4]. Next, let f ∈ C N ([a, b]), where N = μ , (· is the ceiling of the number), 0 0 ( j = 1, ..., m) in terms of a multiple series of the form: E

(γ j ) (γ ,...,γ ) , ..., z m ) = E (ρ11 ,...,ρmm ),λ (z 1 , ..., z m ) = (ρ j ),λ (z 1 ∞ 

(γ1 )k1 ... (γm )km z 1k1 ...z mkm   , m k1 !...km !  k1 ,...,km =0  λ+ kjρj

(2.30)

j=1

  where γ j k j is the Pochhammer symbol,  is the gamma function. By [18], p. 157,   (2.30) converges for Re ρ j > 0, j = 1, ..., m. (γ ,...,γ )

1 m In what follows we will use the particular case of E (ρ,...,ρ),λ [ω1 t ρ , ..., ωm t ρ ], (γ j ) denoted by E (ρ),λ [ω1 t ρ , ..., ωm t ρ ], where 0 < ρ < 1, t ≥ 0, λ > 0, γ j ∈ R with       γ j k j := γ j γ j + 1 ... γ j + k j − 1 , ω j ∈ R − {0}, for j = 1, ..., m.

Let now f ∈ C n+1 ([a, b]), n ∈ Z+ . We define the Caputo type generalized left fractional derivative with non-singular kernel of order n + ρ, as n+ρ n+ρ,λ f (x) := C(γAj )(ω j ) Da∗ f (x) := Da∗

A (ρ) 1−ρ

x a



−ωm ρ (γ j ) −ω1 ρ E (ρ),λ (x − t)ρ , ..., (x − t)ρ f (n+1) (t) dt, 1−ρ 1−ρ

(2.31)

∀ x ∈ [a, b] . Similarly, we define the Caputo type generalized right fractional derivative with non-singular kernel of order n + ρ, as n+ρ

n+ρ,λ

Db− f (x) := C(γAj )(ω j ) Db−

f (x) :=

30

2 Advanced Abstract Fractional Monotone Approximation

(−1)

n+1

A (ρ) 1−ρ

b x



−ωm ρ (γ j ) −ω1 ρ E (ρ),λ (t − x)ρ , ..., (t − x)ρ f (n+1) (t) dt, 1−ρ 1−ρ

(2.32) ∀ x ∈ [a, b] . Above A (ρ) is a normalizing constant. The above derivatives (2.31), (2.32) generalize the Atangana-Baleanu fractional derivatives [8]. We rewrite (2.31) and (2.32), and for [a, b] = [−1, 1] . Let μ > 0 with μ ∈ / N and μ = n ∈ N. That is 0 < 1 − n + μ < 1, and let f ∈ C n ([−1, 1]). Then, we have μ

μ,λ

D−1∗ f (x) := C(γAj )(ω j ) D−1∗ f (x) := A (1 − n + μ) n−μ

x

(γ j )

E (1−n+μ),λ −1



−ω1 (1 − n + μ) (x − t)1−n+μ , ..., n−μ

(2.33)

−ωm (1 − n + μ) 1−n+μ f (n) (t) dt, (x − t) n−μ and

μ μ,λ f (x) := C(γAj )(ω j ) D1− f (x) := D1−

A (1 − n + μ) (−1) n−μ

1

n

x

−ω1 (1 − n + μ) (γ j ) E (1−n+μ),λ (t − x)1−n+μ , ..., (2.34) n−μ

−ωm (1 − n + μ) 1−n+μ f (n) (t) dt, (t − x) n−μ ∀ x ∈ [−1, 1] . m m 0 0 f = f , D1− f = f , and D−1∗ f = f (m) , D1− f = (−1)m f (m) , We will set D−1∗ when m ∈ N. We make Remark 2.7 Fractional Calculi of Sects. 2.2.2 and 2.2.3 are special cases of abstract fractional calculus, see Sect. 2.2.1. In particular the important condition (2.21) is fulfilled. 1 So, we will verify 0 kh (z) dz ≥ 1, h = 0. (I) First for Sect. 2.2.2: We notice that

2.2 Fractional Calculi

31

1

−γ

z N −μ−1 E ρ,N −μ (ωz ρ ) dz =

0

(here ρ, N − μ > 0, γ < 0, ω > 0) 1

∞ 

z N −μ−1

k=0

0 ∞  k=0

∞  k=0 ∞  k=0

(−γ )k (ωz ρ )k dz k! (ρk + N − μ) 1

(−γ )k ωk k! (ρk + N − μ)

(−γ )k ωk k! (ρk + N − μ)

(by [13], p. 175)

=

z N −μ−1 z ρk dz =

(2.35)

0

1

z (ρk+N )−μ−1 dz =

0

(−γ )k ωk −γ = E ρ,N −μ+1 (ω) ≥ 1, k! (ρk + N − μ + 1)

for suitable ω > 0. (II) Next, for Sect. 2.2.3: Here γ j > 0, j = 1, ..., m; λ = 1; N μ > 0, μ = n ∈ N, ω j < 0, j = 1, ..., m. Without loss of generality we assume that A (1 − n + μ) > 0. We have that A (1 − n + μ) n−μ ...,

1 0

−ω1 (1 − n + μ) 1−n+μ (γ j ) z E (1−n+μ),1 , n−μ

−ωm (1 − n + μ) 1−n+μ dz = z n−μ

(here 0 < 1 − (n − μ) = 1 − n + μ < 1) ⎡ A (1 − n + μ) n−μ

1 ⎢  ⎢ ∞ ⎢ ⎢ ⎣k1 ,...,km =0 0

(γ1 ) ... (γm )km  k1   m   1+ k j (1 − n + μ) 

j=1

32

2 Advanced Abstract Fractional Monotone Approximation

m





j=1

−ω j (1−n+μ) n−μ

k j

z

(1−n+μ)

m 

kj

⎤

j=1

k1 !...km !  =

A (1 − n + μ) n−μ

∞ 

(γ1 ) ... (γm )km   k1   m  k1 ,...,km =0  1+ k j (1 − n + μ)

⎥ ⎥ ⎥ ⎥ dz ⎥ ⎦



m  −ω j (1−n+μ) k j 1 j=1

n−μ

k1 !...km !

z

(1−n+μ)

m 

kj

j=1

dz

0

j=1

 =

A (1 − n + μ) n−μ



∞ 

(γ1 )k1 ... (γm )km    m  k1 ,...,km =0  2+ k j (1 − n + μ) 

m j=1



(2.36)

 −ω j (1−n+μ) k j n−μ

k1 !...km !

j=1

A (1 − n + μ) (γ j ) = E (1−n+μ),2 n−μ



−ω1 (1 − n + μ) −ωm (1 − n + μ) , ..., n−μ n−μ

 ≥ 1,

(2.37)

for suitable ω j < 0, for j = 1, ..., m. We also need Definition 2.8 Let f ∈ C n ([−1, 1]), N μ > 0, μ = n ∈ N; ω < 0. That is 0 < 1 − n + μ < 1. The parametrized Caputo-Fabrizio non-singular kernel fractional derivatives, left and right of order μ, respectively, are given as follows (see also [14]): CF μ ω D−1+

CF μ ω D1−

1 f (x) := n−μ

(−1)n f (x) := n−μ

x −1

1 x

  (1 − n + μ) ω exp − (x − t) f (n) (t) dt, n−μ 

 (1 − n + μ) ω exp − (t − x) f (n) (t) dt, n−μ

∀ x ∈ [−1, 1] . Equations (2.38), (2.39) are special cases of (2.17), (2.18). We make Remark 2.9 We want to calculate

(2.38)

(2.39)

2.2 Fractional Calculi

33

1 ∞> 0

  (1 − n + μ) ω exp − z dz n−μ

) (set δ := − (1−n+μ)ω n−μ 1 = 0

∞ 

eδ

 1 1 1 δ e −1 = − = eδz dz = eδz |10 = δ δ δ δ

=

k=0

δ

δk k!

−

1 δ

(2.40)

⎛ ⎞ k−1 1−n+μ k−1 ∞ (−ω)  ⎜ n−μ ⎟ = ⎝ ⎠ ≥ 1, k! k!

∞ k−1  δ k=0

k=0

for suitable ω < 0. So, again condition (2.21) is valid.

2.3 Main Results We present Theorem 2.10 Let h, v, r be integers, 0 ≤ h ≤v ≤ r and let f ∈ C r ([−1, 1]) , with f (r ) having modulus of smoothness ωs f (r ) , δ there, s ≥ 1. Let α j (x), j = h, h + 1, ..., v be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. Let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < αj αr < r . Here k j D∗−1 f stands for the abstract left Caputo fractional derivative of f of order α j anchored at −1. We consider the abstract left fractional linear differential operator v % &  αj , L ∗ := α j (x) k j D∗−1 (2.41) j=h

and suppose, throughout [0, 1],

Set

(2.42)

k j (z) dz, j = 1, ..., r,

(2.43)

2 λ j := 0

and λ0 = 1; along with

L ∗ ( f ) ≥ 0.

34

2 Advanced Abstract Fractional Monotone Approximation

    sup αh−1 (x) α j (x) , j = h, ..., k.

s j :=

(2.44)

−1≤x≤1

Then, for any n ∈ N : n ≥ max (4 (r + 1) , r + s), there exists a real polynomial Q n (x) of degree ≤ n such that L ∗ (Q n ) ≥ 0, throughout [0, 1] ,

(2.45)

and (i)        Cr,s 1 αj αj   f (x) − k j D∗−1 Q n (x) ≤ λ j r − j ωs f (r ) , max  k j D∗−1 , −1≤x≤1 n n

(2.46)

when j = h + 1, ..., r ; (ii) if h = 0 and j = 1, ..., h, we have      αj αj   max  k j D∗−1 f (x) − k j D∗−1 Q n (x) ≤

−1≤x≤1

⎡  ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣

v  j∗ =h

 s j∗ λ j∗ h!

⎤ ⎥   ⎥ αj   max  k j D∗−1 xh  + λ j ⎥ ⎥; −1≤x≤1 ⎦

(2.47)

(iii) furthermore ⎡ max | f (x) − Q n (x)| ≤

−1≤x≤1

 ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣



v 

sjλj

j=h

h!

⎤ ⎥ ⎥ + 1⎥ ⎥. ⎦

(2.48)

Proof Let Q n be as in Corollary 2.2. Let alsoα j > 0, j = 1, ..., r , such that α0 = 0 < α1 < 1 < α2 < 2 < ... < αr < r . That is α j = j, j = 0, 1, ..., r . αj αj f and k j D∗−1 Qn , We consider the abstract left Caputo fractional derivatives k j D∗−1 j = 1, ..., r. We notice that (x ∈ [−1, 1])      αj  kj αj   D∗−1 f (x) − k j D∗−1 Q n (x) =   x   x    k j (x − t) f ( j) (t) dt − k j (x − t) Q (nj) (t) dt  =     −1

−1

(2.49)

2.3 Main Results

35

  x      j) (  k j (x − t) f ( j) (t) − Q n (t) dt  ≤     −1

x

  (2.2.)   ( j) k j (x − t)  f ( j) (t) − Q n (t) dt ≤

−1

⎞ ⎛ x    ⎝ k j (x − t) dt ⎠ Cr,s ωs f (r ) , 1 = nr − j n −1

⎞ ⎞ ⎛ 2 ⎛ x+1       C Cr,s 1 1 r,s (r ) ⎝ ≤ ⎝ k j (z) dz ⎠ r − j ωs f (r ) , . (2.50) k j (z) dz ⎠ r − j ωs f , n n n n 0

0

We have proved that      αj  kj αj   D∗−1 f (x) − k j D∗−1 Q n (x) ≤ ⎛ 2 ⎞    ⎝ k j (z) dz ⎠ Cr,s ωs f (r ) , 1 , nr − j n

(2.51)

0

∀ x ∈ [−1, 1], j = 1, ..., r. Hence it holds      αj αj   f (x) − k j D∗−1 Q n (x) ≤ max  k j D∗−1

−1≤x≤1

Cr,s λ j r − j ωs n where

 f

(r )

(2.52)

 1 , j = 1, ..., r, , n

2 λ j :=

k j (z) dz, j = 1, ..., r, as in ( 2.43.). 0

Inequality (2.52) is valid when j = 0 by (2.2) and (2.9), and we can set λ0 = 1. Set ⎞ ⎛    v 1 ⎝ ρn := Cr,s ωs f (r ) , s j λ j n j−r ⎠ . (2.53) n j=h

I. Suppose, throughout [0, 1], αh (x) ≥ α > 0.

36

2 Advanced Abstract Fractional Monotone Approximation

Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n as in Corollary 2.2, so that       (2.52.) αj αj   max k j D∗−1 f (x) + ρn (h!)−1 x h − k j D∗−1 Q n (x) ≤

−1≤x≤1

λj

Cr,s ωs nr − j



f (r ) ,

(2.54)

 1 , j = 0, 1, ..., r. n

When j = h + 1, ..., r, we get        Cr,s  kj αj  kj αj (r ) 1 D∗−1 Q n (x) ≤ λ j r − j ωs f , , max  D∗−1 f (x) − −1≤x≤1 n n

(2.55)

proving (2.46). For j = 1, ..., h = 0, we have that    (2.54.)   αj αj   f (x) − k j D∗−1 Q n (x) ≤ max  k j D∗−1

−1≤x≤1

  Cr,s ρn αj   x h  + λ j r − j ωs max  k j D∗−1 h! −1≤x≤1 n Cr,s ωs h!



 f

(r )

1 , n

 =

⎞ ⎛     v   Cr,s 1 1 αj   ⎝ f (r ) , s j∗ λ j∗ n j∗ −r ⎠ max  k j D∗−1 x h  + λ j r − j ωs f (r ) , −1≤x≤1 n n n j∗ =h

 = Cr,s ωs

f (r ) ,



⎡

⎛

1 ⎣1 ⎝ n h!

v  j∗ =h

⎡  ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣

⎤(2.56)   λj αj   s j∗ λ j∗ n j∗ −r ⎠ max  k j D∗−1 xh  + r− j ⎦ ≤ −1≤x≤1 n

v  j∗ =h

⎞

 s j∗ λ j∗ h!

⎤ ⎥   ⎥ αj   xh  + λ j ⎥ max  k j D∗−1 ⎥, −1≤x≤1 ⎦

proving (2.47). When j = 0 from (2.54) we get     C 1 r,s   max  f (x) + ρn (h!)−1 x h − Q n (x) ≤ r ωs f (r ) , . −1≤x≤1 n n That is max | f (x) − Q n (x)| ≤

−1≤x≤1

ρn Cr,s + r ωs h! n



f (r ) ,

1 n

(2.57)

 =

(2.58)

2.3 Main Results

37

     C Cr,s ωs f (r ) , n1 v r,s j−r (r ) 1 ≤ + r ωs f , sjλjn j=h h! n n  Cr,s ωs

(r )

f

1 , n



 1 v 1 sjλj + r ≤ j=h h!nr −v n

(2.59)

  ⎤   ⎡ v Cr,s ωs f (r ) , n1 j=h s j λ j ⎣ + 1⎦ , nr −v h! proving (2.48). Also if 0 ≤ x ≤ 1, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) + v 

αh−1 (x) α j (x)

%

kj

ρn h!

kh

αh D∗−1 xh+

(2.60)

& (2.54.) ≥

ρn h!

kj

j D∗−1 xh

  ρ 1 v n , s j λ j n j−r = j=h n h!

kh

h D∗−1 x h − ρn =: ϕ

α

j D∗−1 Q n (x) −

kj

α

j D∗−1 f (x) −

j=h

α

(by L ∗ f ≥ 0) ρn h!

kh

αh D∗−1 xh

 − Cr,s ωs

f

(r )

α

(if h = 0, then αh = 0, and ϕ = 0). If h = 0, then  ϕ = ρn

k h D αh x h ∗−1

h!

⎞ ⎛ x  − 1 = ρn ⎝ kh (x − t) dt − 1⎠ = 

(2.61)

−1

⎞ ⎞ ⎛ x+1 ⎛ 1   ρn ⎝ kh (z) dz − 1⎠ ≥ ρn ⎝ kh (z) dz − 1⎠ ≥ 0, 0

0

1 by the assumption (2.21): 0 kh (z) dz ≥ 1, when h = 0. Hence in both cases we get L ∗ (Q n (x)) ≥ 0, x ∈ [0, 1] .

(2.62)

II. Suppose on [0, 1] that αh (x) ≤ β < 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n as in Corollary 2.2, such that

38

2 Advanced Abstract Fractional Monotone Approximation

      (2.52.) αj αj   f (x) − ρn (h!)−1 x h − k j D∗−1 max k j D∗−1 Q n (x) ≤

(2.63)

−1≤x≤1

λj

Cr,s ωs nr − j



f (r ) ,

 1 , j = 0, 1, ..., r. n

Similarly, we obtain again the inequalities of convergence, see (2.46), (2.47) and (2.48). Also if 0 ≤ x ≤ 1, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) − v 

αh−1 (x) α j (x)

%

kj

α

j D∗−1 Q n (x) −

kj

ρn h!

α

j D∗−1 f (x) +

j=h

kh

ρn h!

αh D∗−1 xh+

kj

α

j D∗−1 xh

& (2.63.) ≤

(by L ∗ f ≥ 0) ρn − h!

kh

αh D∗−1 x h + Cr,s ωs



⎞ ⎛   v 1 ρn ⎝ s j λ j n j−r ⎠ = − f (r ) , n h!

kh

αh D∗−1 x h + ρn =: ψ

j=h

(2.64) (if h = 0, then αh = 0, and ψ = 0). If h = 0, then  ψ = ρn 1 −

k h D αh x h ∗−1

h!



⎡ = ρn ⎣1 −

x

⎤ kh (x − t) dt ⎦ =

−1

⎤ ⎤ ⎡ x+1 1 ρn ⎣1 − kh (z) dz ⎦ ≤ ρn ⎣1 − kh (z) dz ⎦ ≤ 0. ⎡

0

0

Hence again in both cases it holds L ∗ (Q n (x)) ≥ 0, ∀ x ∈ [0, 1] .

(2.65)

We also give Theorem 2.11 Let h, v, r be integers, h is even, 0 ≤ h ≤ v ≤ r and let f ∈ C r ([−1, 1]) , with f (r ) having modulus of smoothness ωs f (r ) , δ there, s ≥ 1. Let α j (x), j = h, h + 1, ..., v be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [−1, 0]. Let the real numbers α0 = 0 < α1 < 1 < α α2 < 2 < ... < αr < r . Here k j D1−j f stands for the abstract right Caputo fractional derivative of f of order α j anchored at 1. We consider the abstract right fractional linear differential operator

2.3 Main Results

39

L ∗ :=

v 

α j (x)

%

kj

& α D1−j ,

(2.66)

j=h

and suppose, throughout [−1, 0],

L ∗ ( f ) ≥ 0.

(2.67)

k j (z) dz, j = 1, ..., r,

(2.68)

    sup αh−1 (x) α j (x) , j = h, ..., k.

(2.69)

Set

2 λ j := 0

and λ0 = 1; along with s j :=

−1≤x≤1

Then, for any n ∈ N : n ≥ max (4 (r + 1) , r + s), there exists a real polynomial Q n (x) of degree ≤ n such that L ∗ (Q n ) ≥ 0, throughout [−1, 0] ,

(2.70)

and (i)        Cr,s   kj αj kj αj (r ) 1 , D1− Q n (x) ≤ λ j r − j ωs f , max  D1− f (x) − −1≤x≤1 n n

(2.71)

when j = h + 1, ..., r ; (ii) if h = 0 and j = 1, ..., h, we have      α α   max  k j D1−j f (x) − k j D1−j Q n (x) ≤

−1≤x≤1

⎡  ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣



v 

s j∗ λ j∗

j∗ =h

h!

⎤ ⎥   ⎥ α   max  k j D1−j x h  + λ j ⎥ ⎥; −1≤x≤1 ⎦

(2.72)

(iii) furthermore ⎡ max | f (x) − Q n (x)| ≤

−1≤x≤1

 ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣



v 

sjλj

j=h

h!

⎤ ⎥ ⎥ + 1⎥ ⎥. ⎦

(2.73)

40

2 Advanced Abstract Fractional Monotone Approximation

Proof Let Q n be as in Corollary 2.2. Let alsoα j > 0, j = 1, ..., r , such that α0 = 0 < α1 < 1 < α2 < 2 < ... < αr < r . That is α j = j, j = 0, 1, ..., r . α α We consider the abstract right Caputo fractional derivatives k j D1−j f and k j D1−j Q n , j = 1, ..., r. We notice that (x ∈ [−1, 1])      αj  kj αj   D1− f (x) − k j D1− Q n (x) =     1 1   j) ( j) j j ( = (−1) − x) f dt − − x) Q dt k k (t (t) (−1) (t (t) j j n     x

(2.74)

x

  1      j) (  k j (t − x) f ( j) (t) − Q n (t) dt  ≤     x

1

  (2.2.)   ( j) k j (t − x)  f ( j) (t) − Q n (t) dt ≤

x

⎛ ⎝

⎞

1

Cr,s k j (t − x)⎠ r − j ωs n



f (r ) ,

1 n

 =

x

⎛ 2 ⎞ ⎞ ⎛ 1−x       C Cr,s 1 1 r,s ⎝ ≤ ⎝ k j (z) dz ⎠ r − j ωs f (r ) , . (2.75) k j (z) dz ⎠ r − j ωs f (r ) , n n n n 0

We have proved that

0

     αj   kj αj  D1− f (x) − k j D1− Q n (x) ≤ ⎛ 2 ⎞    ⎝ k j (z) dz ⎠ Cr,s ωs f (r ) , 1 , nr − j n

(2.76)

0

∀ x ∈ [−1, 1], j = 1, ..., r. Hence it holds      α α   max  k j D1−j f (x) − k j D1−j Q n (x) ≤

−1≤x≤1

λj

Cr,s ωs nr − j



f (r ) ,

 1 , j = 1, ..., r, n

(2.77)

2.3 Main Results

41

where

2 k j (z) dz, j = 1, ..., r, as in (2.68).

λ j := 0

Inequality (2.77) is valid when j = 0 by (2.2) and (2.9), and we can set λ0 = 1. Set ⎞ ⎛    v j−r ⎠ (r ) 1 ⎝ ρn := Cr,s ωs f , sjλjn (2.78) . n j=h

I. Suppose, throughout [−1, 0], αh (x) ≥ α > 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n as in Corollary 2.2, so that       (2.77.) α α   max k j D1−j f (x) + ρn (h!)−1 x h − k j D1−j Q n (x) ≤ (2.79) −1≤x≤1

λj

Cr,s ωs nr − j



f (r ) ,

 1 , j = 0, 1, ..., r. n

When j = h + 1, ..., r, we get        Cr,s 1 α α   , max  k j D1−j f (x) − k j D1−j Q n (x) ≤ λ j r − j ωs f (r ) , −1≤x≤1 n n

(2.80)

proving (2.71). For j = 1, ..., h = 0, we have that    (2.79.)   α α   max  k j D1−j f (x) − k j D1−j Q n (x) ≤

−1≤x≤1

  Cr,s ρn α   max  k j D1−j x h  + λ j r − j ωs h! −1≤x≤1 n Cr,s ωs h!

 f

(r )



f (r ) ,

1 n

 =

⎛ ⎞    v   Cr,s 1 ⎝ 1 α   j∗ −r ⎠ max  k j D1−j x h  + λ j r − j ωs f (r ) , , s j∗ λ j∗ n −1≤x≤1 n n n j∗ =h

 = Cr,s ωs

f (r ) ,



⎡

⎛

1 ⎣1 ⎝ n h!

v  j∗ =h

⎡  ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣

⎤(2.81)   λj α   s j∗ λ j∗ n j∗ −r ⎠ max  k j D1−j x h  + r − j ⎦ ≤ −1≤x≤1 n

v  j∗ =h

⎞

 s j∗ λ j∗ h!

⎤ ⎥   ⎥ α   max  k j D1−j x h  + λ j ⎥ ⎥, −1≤x≤1 ⎦

42

2 Advanced Abstract Fractional Monotone Approximation

proving (2.72). When j = 0 from (2.79) we get     C 1 r,s   . max  f (x) + ρn (h!)−1 x h − Q n (x) ≤ r ωs f (r ) , −1≤x≤1 n n That is max | f (x) − Q n (x)| ≤

−1≤x≤1

ρn Cr,s + r ωs h! n



1 n

f (r ) ,

(2.82)

 =

⎞  ⎛ v    Cr,s ωs f (r ) , n1 Cr,s 1 j−r ⎝ ≤ s j λ j n ⎠ + r ωs f (r ) , h! n n

(2.83)

j=h

 Cr,s ωs

⎛ ⎞ ⎤ ⎡  v  1 1 1 ⎝ ⎣ f (r ) , sjλj⎠ + r ⎦ ≤ n h!nr −v n j=h

⎡  ⎢ Cr,s ωs f (r ) , n1 ⎢ ⎢ ⎢ nr −v ⎣



v 

sjλj

j=h

h!

⎤ ⎥ ⎥ + 1⎥ ⎥, ⎦

(2.84)

proving (2.73). Also if −1 ≤ x ≤ 0, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) + v 

αh−1 (x) α j (x)

%

kj

α

D1−j Q n (x) −

kj

α

ρn h!

D1−j f (x) −

j=h

ρn h!

kh

αh h D1− x +

kj

α

D1−j x h

(2.85)

& (2.79.) ≥

(by L ∗ f ≥ 0) ρn h!

k h αh h D1− x

 − Cr,s ωs

⎞ ⎛   v ρn 1 ⎝ f (r ) , s j λ j n j−r ⎠ = n h!

kh

α

D1−h x h − ρn =: ϕ

j=h

(if h = 0, then αh = 0, and ϕ = 0). If h = 0, then  ϕ = ρn

k h D αh x h 1−

h!



⎛

− 1 = ρn ⎝(−1)h

1 x

⎞ kh (t − x) dt − 1⎠

(h is even)

=

(2.86)

2.3 Main Results

43

⎛ ρn ⎝

1

⎞

⎞ ⎛ 1−x  kh (t − x) dt − 1⎠ = ρn ⎝ kh (z) dz − 1⎠ ≥

x

0

⎛ ρn ⎝

1

⎞ kh (z) dz − 1⎠ ≥ 0,

0

1 by the assumption (2.21): 0 kh (z) dz ≥ 1, when h = 0. Hence in both cases we get L ∗ (Q n (x)) ≥ 0, x ∈ [−1, 0] .

(2.87)

II. Suppose on [−1, 0] that αh (x) ≤ β < 0. Let Q n (x), x ∈ [−1, 1], be a real polynomial of degree ≤ n as in Corollary 2.2, so that       (2.77.) α α   max k j D1−j f (x) − ρn (h!)−1 x h − k j D1−j Q n (x) ≤ (2.88) −1≤x≤1

λj

Cr,s ωs nr − j



f (r ) ,

 1 , j = 0, 1, ..., r. n

Similarly, we obtain again the inequalities of convergence, see (2.71), (2.72) and (2.73). Also if −1 ≤ x ≤ 0, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) − v 

αh−1 (x) α j (x)

%

kj

α

D1−j Q n (x) −

kj

α

ρn h!

D1−j f (x) +

j=h

ρn h!

kh

αh h D1− x +

kj

α

D1−j x h

& (2.88.) ≤

(by L ∗ f ≥ 0) ρn − h!

kh

αh h D1− x + Cr,s ωs



⎞ ⎛   v 1 ρn ⎝ s j λ j n j−r ⎠ = − f (r ) , n h!

kh

αh h D1− x + ρn =: ψ

j=h

(2.89) (if h = 0, then αh = 0, and ψ = 0). If h = 0, then  ψ = ρn 1 −

k h D αh x h 1−

h!



⎡ = ρn ⎣1 −

1 x

⎤ kh (t − x) dt ⎦ =

44

2 Advanced Abstract Fractional Monotone Approximation

 ρn 1 −

1−x



⎡

kh (z) dz ≤ ρn ⎣1 −

0

1

⎤ kh (z) dz ⎦ ≤ 0.

0

Hence again in both cases it holds L ∗ (Q n (x)) ≥ 0, ∀ x ∈ [−1, 0] .

(2.90)

Comment 2.12 Clearly Theorem 2.10 generalizes Theorem 2.5, and Theorem 2.11 generalizes Theorem 4.4, p. 35 of [3]. Furthermore there, the approximating polynomial Q n depends on f, ρn , h; which ρn depends on n, Cr,s , n, v, s j , λ j ; which λ j depends on k j . I.e. polynomial Q n among others depends on the type of fractional calculus we use. Consequently, Theorem 2.10 is valid for the following left fractional linear differential operators: (1) v % &  γ L ∗1 := α j (x) C Dρ,α j ,ω,−1+ , (2.91) j=h

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (2.26)); (2) v % α &  j L ∗2 := , α j (x) D−1∗ (2.92) j=h

(see (2.33)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and (3) v % &  F αj L ∗3 := α j (x) C (2.93) ω D−1+ , j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Similarly, Theorem 2.11 is valid for the following right fractional linear differential operators: (1)∗ v % &  γ L ∗∗ := α j (x) C Dρ,α j ,ω,1− , (2.94) 1 j=h

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (2.27)); (2)∗ v % α &  L ∗∗ := α j (x) D1−j , (2.95) 2 j=h

2.3 Main Results

45

(see (2.34)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and (3)∗ v % &  F αj L ∗∗ α j (x) C (2.96) ω D1− , 3 := j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Our developed abstract simultaneous fractional monotone approximation theory with its applications involves weaker conditions than the one with ordinary derivatives ([2]), and can cover many diverse general cases in a multitude of complex settings and environments.

References 1. Anastassiou, G.A.: Bivariate monotone approximation. Proc. Amer. Math. Soc. 112(4), 959– 964 (1991) 2. Anastassiou, G.A.: High order monotone approximation with linear differential operators. Indian J. Pure Appl. Math. 24(4), 263–266 (1993) 3. Anastassiou, G.A.: Frontiers in Approximation Theory. World Scientific Publishing Co Pte Ltd., New Jersey, Singapore (2015) 4. Anastassiou, G.A.: Foundations of Generalized Prabhakar-Hilfer Fractional Calculus with Applications. Cubo (2021, accepted) 5. Anastassiou, G.A.: Multiparameter fractional differentiation with non singular kernel. Issues Anal. (2021, accepted) 6. Anastassiou, G.A.: Univariate simultaneous high order abstract fractional monotone approximation with applications (2021, submitted) 7. Anastassiou, G.A., Shisha, O.: Monotone approximation with linear differential operators. J. Approx. Theor. 44, 391–393 (1985) 8. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) 9. Diethelm, K.: The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, vol. 2004, 1st ed. Springer, New York, Heidelberg (2010) 10. Giusti, A., et al.: A practical guide to Prabhakar fractional calculus. Fract. Calc. Appl. Anal. 23(1), 9–54 (2020) 11. Gonska, H.H., Hinnermann, E.: Pointwise estimated for approximation by algebraic polynomials. Acta Math. Hungar. 46, 243–254 (1985) 12. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler functions. Relat. Top. Appl., Springer, Heidelberg, New York (2014) 13. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York, Heidelberg, Berlin (1965) 14. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1 (2), 87–92 (2015) 15. Polito, F., Tomovski, Z.: Some properties of Prabhakar-type fractional calculus operators. Fract. Differ. Calc. 6(1), 73–94 (2016) 16. Saxena, R.K., Kalla, S.L., Saxena, R.: Integr. Transforms Spec. Functi. Multivariate analogue of generalized Mittag-Leffler function 22(7), 533–548 (2011)

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2 Advanced Abstract Fractional Monotone Approximation

17. Shisha, O.: Monotone approximation. Pacific J. Math. 15, 667–671 (1965) 18. Srivastava, H.M., Daoust, M.C.: A note on the convergence of Kompe’ de Feriet’s double hypergeometric series. Math. Nachr. 53, 151–159 (1972)

Chapter 3

Spline Abstract Fractional Monotone Approximation

Here we extend our earlier high order simultaneous fractional polynomial spline monotone approximation theory to abstract high order simultaneous fractional polynomial spline monotone approximation, with applications to Prabhakar fractional calculus and non-singular kernel fractional calculi. We cover both the left and right sides of this constrained approximation. Let f ∈ C s ([−1, 1]), s ∈ N and L ∗ be a linear left or right side fractional differential operator such that L ∗ ( f ) ≥ 0 over [0, 1] or [−1, 0], respectively. Then there exists a sequence Q n , n ∈ N of polynomial splines with equally spaced knots of given fixed order such that L ∗ (Q n ) ≥ 0 on [0, 1] or [−1, 0], respectively. Furthermore f is approximated with rates fractionally and simultaneously by Q n in th uniform norm. This constrained fractional approximation on [−1, 1] is given via inequalities involving a higher modulus of smoothness of f (s) . It follows [5].

3.1 Introduction Let [a, b] ⊂R and with points  for n ≥ 1 consider the partition   n := n ([a,  b])b−a , i = 0, 1, ..., n. Hence x =  ≡ max − x . xin = a + i b−a n in i−1,n n n 1≤i≤n

Let Sm (n ) be the space of polynomial splines of order m > 0 with simple knots at the points xin , i = 1, ..., n − 1 (see [15, p. 5]). Then there exists a linear operator Q n : Q n ≡ Q n ( f ), mapping B [a, b]: the space of bounded real valued functions f on [a, b], into Sm (n ) (see [15, p. 224], Theorem 6.18). From the same reference [15, p. 227], Corollary 6.21, we get

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_3

47

48

3 Spline Abstract Fractional Monotone Approximation

Corollary 3.1 Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ C σ −1 ([a, b]); r = 0, ..., σ − 1,   (r )  f − Q (r )  ≤ C1



n

b−a n

σ −r −1

 ωm−σ +1

f

(σ −1)

b−a , n

 ,

(3.1)

where C1 depends only on m, C1 = C1 (m). By denoting C2 = C1 max (b − a)σ −r −1 we obtain 0≤r ≤σ −1

Lemma 3.2 ([1]) Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ C σ −1 ([a, b]); r = 0, ..., σ − 1,    (r )   f − Q (r )  ≤ C2 ωm−σ +1 f (σ −1) , b − a , (3.2) n ∞ n σ −r −1 n where C2 depends only on m, σ and b − a. Here ωm−σ +1 is the usual modulus of smoothness of order m − σ + 1. We are motivated by Theorem 3.3 ([1]) Let h, k, σ, m be integers, 0 ≤ h ≤ k ≤ σ − 1, σ ≤ m and let f ∈ C σ −1 [a, b]. Let α j (x) ∈ B [a, b], j = h, h + 1, ..., k and suppose that αh (x) ≥ α > 0 or αh (x) ≤ β < 0 for all x ∈ [a, b]. Take the linear differential operator  j k  d (3.3) L= α j (x) d xj j=h and assume, throughout [a, b] ,

L ( f ) ≥ 0.

(3.4)

Then, for every integer n ≥ 1, there  a polynomial spline  function Q n (x) of  is , i = 1, ..., n − 1 such that L (Q n ) ≥ 0 order m with simple knots at a + i b−a n throughout [a, b] ,and   (r )  f − Q (r ) 

∞

n

≤



C

n

ω σ −k−1 m−σ +1

f (σ −1) ,

b−a n

 , 0 ≤ r ≤ h.

(3.5)

Moreover, we find   (r )  f − Q (r )  n

∞

≤

C n σ −r −1

 ωm−σ +1

f

(σ −1)

b−a , n

 , h + 1 ≤ r ≤ σ − 1, (3.6)

where C is a constant independent of f and n. It depends only on m, σ, L , a, b.

3.1 Introduction

49

Next we specialize on the case of a = −1, b = 1. That is working on [−1, 1]. By Lemma 3.2 we get Lemma 3.4 Let 1 ≤ σ ≤ m, n ≥ 1. Then for all f ∈ C σ −1 ([−1, 1]); j = 0, 1, ..., σ − 1,    ( j)   f − Q ( j)  ≤ C2 ωm−σ +1 f (σ −1) , 2 , (3.7) n ∞ n σ − j−1 n where C2 := C2 (m, σ ) := C1 (m) 2σ −1 . 

Since ωm−σ +1

f (σ −1) ,

2 n



≤ 2m−σ +1 ωm−σ +1



f (σ −1) ,

1 n

 (3.8)

(see [7, p. 45]), we obtain Lemma 3.5 Let 1 ≤ σ ≤ m, n ≥ 1. Then, for each f ∈ C σ −1 ([−1, 1]); j = 0, 1, ..., σ − 1, there exists Q n := Q n ( f ) ∈ Sm (n ([−1, 1])) such that  ( j)   f − Q ( j)  n

∞

≤



C2∗

n

ω σ − j−1 m−σ +1

f (σ −1) ,

 1 , n

(3.9)

where C2∗ := C2∗ (m, σ ) := C1 (m) 2m . We use a lot Lemma 3.5 in this chapter. In [2], chap. 5, we extended Theorem 3.3 over [−1, 1] to the fractional level. Indeed there L is replaced by L ∗ , a linear left Caputo fractional differential operator. Then the monotonicity property is only true on the critical interval [0, 1]. Simultaneous fractional convergence remained true on all of [−1, 1]. We make Definition 3.6 ([8, p. 50]) Let α > 0 and α = m, ( · ceiling of the number). Consider f ∈ C m ([−1, 1]). We define the left Caputo fractional derivative of f of order α as follows: 

α D∗−1



1 f (x) =  (m − α)

x

(x − t)m−α−1 f (m) (t) dt,

−1

for any x ∈ [−1, 1], where  is the gamma function. We set 0 f (x) = f (x) , D∗−1 m f (x) = f (m) (x) , ∀ x ∈ [−1, 1] . D∗−1

So in [2], chap. 5, we proved

(3.10)

(3.11)

50

3 Spline Abstract Fractional Monotone Approximation

Theorem 3.7 Let h, k, σ, m be integers, 1 ≤ σ ≤ m, n ∈ N, with 0 ≤ h ≤ k ≤ σ − 2 and f ∈ C σ −1 ([−1, 1]), with f (σ −1) having modulus of smoothness  (σlet −1) , δ there, δ > 0. Let α j (x), j = h, h + 1, ..., k be real functions, ωm−σ +1 f defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. Let the real numbers α0 = 0 < α1 ≤ 1 < α2 ≤ 2 < ... < ασ −2 ≤ σ − 2. αj f stands for the left Caputo fractional derivative of f of order α j anchored Here D∗−1 at −1. Consider the linear left fractional differential operator L ∗ :=

k 

αj  α j (x) D∗−1

(3.12)

j=h

and suppose, throughout [0, 1] , L ∗ ( f ) ≥ 0. Then, for every integer n ≥ 1, there exists a polynomial spline function Q n (x) of  order m > 0 with simple knots at −1 + i n2 , i = 1, ..., n − 1 such that L ∗ (Q n ) ≥ 0 throughout [0, 1] ,and  α j     αj f (x) − D∗−1 Q n (x) ≤ sup  D∗−1

−1≤x≤1

2 j−α j C∗   σ −2j−1 ωm−σ +1  j − αj + 1 n

 f

(σ −1)

 1 , , n

(3.13)

j = h + 1, ..., σ − 2. Set   l j :≡ sup αh−1 (x) α j (x) , h ≤ j ≤ k.

(3.14)

x∈[−1,1]

When j = 1, ..., h we derive  α j   αj   max  D∗−1 f (x) − D∗−1 Q n (x) ≤

−1≤x≤1

 k 

2τ −ατ lτ  (τ − ατ + 1) τ =h

  h− j  λ=0



C2∗

n

ω σ −k−1 m−σ +1

2h−α j −λ   λ! h − α j − λ + 1



f (σ −1) ,

 1 · n

 2 j−α j  . +   j − αj + 1 (3.15)

Finally it holds sup | f (x) − Q n (x)| ≤

−1≤x≤1

C2∗ n σ −k−1

 ωm−σ +1

f

(σ −1)

1 , n



 k 1  2τ −ατ +1 . lτ h! τ =h  (τ − ατ + 1)

(3.16)

In this chapter we establish abstract fractional calculus left and right high order monotone approximation theory (by polynomial splines of simple knots) of Caputo

3.1 Introduction

51

type, and then we apply our results to Prabhakar fractional Calculus, generalized nonsingular fractional calculus, and parametrized Caputo-Fabrizio non-singular fractional calculus. Next, we build the related necessary fractional calculi background.

3.2 Fractional Calculi Here we need to be very specific in preparation for our main results.

3.2.1 Abstract Fractional Calculus Let h, v, σ, m be non-negative integers, 2 ≤ σ ≤ m, with 0 ≤ h ≤ v ≤ σ − 2. Let also N α j > 0, j = 1, ..., σ − 2, such  that α0 = 0 < α1 < 1 < α2 < 2 < α3 < 3 < ... < ... < ασ −2 < σ − 2. That is α j = j, j = 1, ..., σ − 2; α0 = 0. Consider the integrable functions k j := K α j : [0, 2] → R+ , j = 0, 1, ..., σ − 2. Here g ∈ C σ −1 ([−1, 1]) . We consider the following abstract left side Caputo type fractional derivatives: k j

 αj D∗−1 g (x)

x :=

k j (x − t) g ( j) (t) dt,

(3.17)

−1

j = 1, .., σ − 2; ∀ x ∈ [−1, 1] . Similarly, we define the corresponding right side generalized Caputo type fractional derivatives: k j

 α D1−j g (x) := (−1) j

1

k j (t − x) g ( j) (t) dt,

(3.18)

x

j = 1, .., σ − 2; ∀ x ∈ [−1, 1] . We set     j j kj D∗−1 g (x) := g ( j) (x) ; k j D1− g (x) := (−1) j g ( j) (x) ,

(3.19)

for j = 1, ..., σ − 2, and also we set k0 ∀ x ∈ [−1, 1] .

   0 0 D∗−1 g (x) := k0 D1− g (x) := g (x) ,

(3.20)

52

3 Spline Abstract Fractional Monotone Approximation

We will assume that 1 kh (z) dz ≥ 1, when h = 0.

(3.21)

0

In the usual Caputo fractional derivatives case it is k j (z) =

z j−α j −1  , j = 1, ..., σ − 2; ∀ z ∈ [0, 2] ,   j − αj

(3.22)

and (3.21) is fulfilled, by the fact that  (h − αh + 1) ≤ 1, see [2, p. 6].

3.2.2 About Prabhakar Fractional Calculus Here we follow [3, 13]. We consider the Prabhakar function (also known as the three parameter MittagLeffler function), (see [10, p. 97]; [9]) γ

E α,β (z) =

∞  k=0

(γ )k zk , k! (αk + β)

(3.23)

where  is the gamma function; α, β > 0, γ ∈ R, z ∈ R, and 1 0 . (z) = (β) (γ )k = γ (γ + 1) ... (γ + k − 1). It is E α,β Let a, b ∈ R, a < b and x ∈ [a, b]; f ∈ C ([a, b]) . The left and right Prabhakar fractional integrals are defined ([3, 13]) as follows:  γ  eρ,μ,ω,a+ f (x) =

x

 γ ω (x − t)ρ f (t) dt, (x − t)μ−1 E ρ,μ

(3.24)

b  

 γ γ eρ,μ,ω,b− f (x) = (t − x)μ−1 E ρ,μ ω (t − x)ρ f (t) dt,

(3.25)

a

and

x

where ρ, μ > 0; γ , ω ∈ R. Functions (3.24) and (3.25) are continuous, see [3]. Next, let f ∈ C N ([a, b]), where N = μ , ( · is the ceiling of the number), 0 0 ( j = 1, ..., m) in terms of a multiple series of the form: E

(γ j ) (γ ,...,γ ) , ..., z m ) = E (ρ11 ,...,ρmm ),λ (z 1 , ..., z m ) = (ρ j ),λ (z 1 ∞ 

(γ1 )k1 ... (γm )km z 1k1 ...z mkm   , m k1 !...km !  k1 ,...,km =0  λ+ kjρj

(3.30)

j=1

  where γ j k j is the Pochhammer symbol,  is the gamma function. By [16, p. 157],   (3.30) converges for Re ρ j > 0, j = 1, ..., m.

54

3 Spline Abstract Fractional Monotone Approximation (γ ,...,γ )

1 m In what follows we will use the particular case of E (ρ,...,ρ),λ [ω1 t ρ , ..., ωm t ρ ], (γ j ) denoted by E (ρ),λ [ω1 t ρ , ..., ωm t ρ ], where 0 < ρ < 1, t ≥ 0, λ > 0, γ j ∈ R with       γ j k j := γ j γ j + 1 ... γ j + k j − 1 , ω j ∈ R − {0}, for j = 1, ..., m.

Let now f ∈ C n+1 ([a, b]), n ∈ Z+ . We define the Caputo type generalized left fractional derivative with non-singular kernel of order n + ρ, as n+ρ n+ρ,λ f (x) := C(γAj )(ω j ) Da∗ f (x) := Da∗

A (ρ) 1−ρ

x a

 −ωm ρ (γ j ) −ω1 ρ ρ ρ E (ρ),λ (x − t) , ..., (x − t) f (n+1) (t) dt, 1−ρ 1−ρ

(3.31)

∀ x ∈ [a, b] . Similarly, we define the Caputo type generalized right fractional derivative with non-singular kernel of order n + ρ, as n+ρ

n+ρ,λ

Db− f (x) := C(γAj )(ω j ) Db− (−1)

n+1

A (ρ) 1−ρ

b x

f (x) :=

 −ωm ρ (γ j ) −ω1 ρ ρ ρ E (ρ),λ (t − x) , ..., (t − x) f (n+1) (t) dt, 1−ρ 1−ρ

(3.32) ∀ x ∈ [a, b] . Above A (ρ) is a normalizing constant. The above derivatives (3.31), (3.32) generalize the Atangana-Baleanu fractional derivatives [6]. We rewrite (3.31) and (3.32), and for [a, b] = [−1, 1] . Let μ > 0 with μ ∈ / N and μ = n ∈ N. That is 0 < 1 − n + μ < 1, and let f ∈ C n ([−1, 1]). Then, we have μ μ,λ f (x) := C(γAj )(ω j ) D−1∗ f (x) := D−1∗

A (1 − n + μ) n−μ

x −1

 −ω1 (1 − n + μ) (γ j ) E (1−n+μ),λ (x − t)1−n+μ , ..., n−μ

−ωm (1 − n + μ) (x − t)1−n+μ f (n) (t) dt, n−μ and

μ

μ,λ

D1− f (x) := C(γAj )(ω j ) D1− f (x) :=

(3.33)

3.2 Fractional Calculi

55

A (1 − n + μ) (−1) n−μ

1

n

x

 −ω1 (1 − n + μ) (γ j ) E (1−n+μ),λ (t − x)1−n+μ , ..., (3.34) n−μ

−ωm (1 − n + μ) 1−n+μ f (n) (t) dt, (t − x) n−μ ∀ x ∈ [−1, 1] . m m 0 0 f = f , D1− f = f , and D−1∗ f = f (m) , D1− f = (−1)m f (m) , We will set D−1∗ when m ∈ N. We make Remark 3.9 Fractional Calculi of Sects. 3.2.2 and 3.2.3 are special cases of abstract fractional calculus, see Sect. 3.2.1. In particular the important condition (3.21) is fulfilled. 1 So, we will verify 0 kh (z) dz ≥ 1, h = 0. (I) First for Sect. 3.2.2: We notice that 1 −γ z N −μ−1 E ρ,N −μ (ωz ρ ) dz = 0

(here ρ, N − μ > 0, γ < 0, ω > 0) 1

∞ 

z N −μ−1

k=0

0 ∞  k=0

∞  k=0 ∞  k=0

(−γ )k (ωz ρ )k dz k! (ρk + N − μ)

(−γ )k ωk k! (ρk + N − μ)

(−γ )k ωk k! (ρk + N − μ)

1

(by [11], p. 175)

=

z N −μ−1 z ρk dz =

(3.35)

0

1

z (ρk+N )−μ−1 dz =

0

(−γ )k ωk −γ = E ρ,N −μ+1 (ω) ≥ 1, k! (ρk + N − μ + 1)

for suitable ω > 0. (II) Next, for Sect. 3.2.3: Here γ j > 0, j = 1, ..., m; λ = 1; N μ > 0, μ = n ∈ N, ω j < 0, j = 1, ..., m. Without loss of generality we assume that A (1 − n + μ) > 0.

56

3 Spline Abstract Fractional Monotone Approximation

We have that A (1 − n + μ) n−μ

1



(γ j )

−ω1 (1 − n + μ) 1−n+μ z , n−μ

E (1−n+μ),1 0

−ωm (1 − n + μ) 1−n+μ dz = z ..., n−μ (here 0 < 1 − (n − μ) = 1 − n + μ < 1) ⎡ A (1 − n + μ) n−μ

1 ⎢  ⎢ ∞ ⎢ ⎢ ⎣k1 ,...,km =0 0

(γ1 ) ... (γm )km  k1   m   1+ k j (1 − n + μ) 

j=1



m 



j=1

−ω j (1−n+μ) n−μ

k j

z

(1−n+μ)

m 

kj

⎤

j=1

k1 !...km !  =

∞ 

A (1 − n + μ) n−μ

⎥ ⎥ ⎥ ⎥ dz ⎥ ⎦



k j m   −ω j (1−n+μ)

(γ1 ) ... (γm )km  k1   m  k1 ,...,km =0  1+ k j (1 − n + μ) 

j=1

n−μ

k1 !...km !

1 z

(1−n+μ)

m  j=1

kj

dz

0

j=1

 =

A (1 − n + μ) n−μ



∞ 

(3.36) k j m   −ω j (1−n+μ)

(γ1 ) ... (γm )km  k1   m  k1 ,...,km =0  2+ k j (1 − n + μ) 

j=1

n−μ

k1 !...km !

j=1

A (1 − n + μ) (γ j ) E (1−n+μ),2 = n−μ



−ω1 (1 − n + μ) −ωm (1 − n + μ) , ..., n−μ n−μ

for suitable ω j < 0, for j = 1, ..., m. We also need

 ≥ 1, (3.37)

3.2 Fractional Calculi

57

Definition 3.10 Let f ∈ C n ([−1, 1]), N μ > 0, μ = n ∈ N; ω < 0. That is 0 < 1 − n + μ < 1. The parametrized Caputo-Fabrizio non-singular kernel fractional derivatives, left and right of order μ, respectively, are given as follows (see also [12]): CF μ ω D−1+

CF μ ω D1−

1 f (x) := n−μ

(−1)n f (x) := n−μ

x −1

1 x

  (1 − n + μ) ω exp − (x − t) f (n) (t) dt, n−μ

(3.38)

  (1 − n + μ) ω exp − (t − x) f (n) (t) dt, n−μ

(3.39)

∀ x ∈ [−1, 1] . Equations (3.38), (3.39) are special cases of (3.17), (3.18). We make Remark 3.11 We want to calculate 1 ∞> 0

  (1 − n + μ) ω z dz exp − n−μ

(set δ := − (1−n+μ)ω ) n−μ 1 =

∞ 

eδz dz =

0

=

 e 1 1 δz 1 1 δ e |0 = e −1 = − = δ δ δ δ δ

δ

−

⎞ ⎛ k−1 1−n+μ k−1 ∞ (−ω)  ⎟ ⎜ n−μ = ⎠ ≥ 1, ⎝ k! k! k=0

∞  δ k−1 k=0

k=0

δk k!

for suitable ω < 0. So, again condition (3.21) is valid.

1 δ

(3.40)

58

3 Spline Abstract Fractional Monotone Approximation

3.3 Main Results We present Theorem 3.12 Let h, v, σ, m be integers, 2 ≤ σ ≤ m, n ∈ N, with 0 ≤ h ≤ v ≤ σ −1 σ − 2 and f ∈ ([−1, 1]) , with f (σ −1) having modulus of smoothness  (σlet  C −1) , δ there, δ > 0. Let α j (x), j = h, h + 1, ..., v be real functions, ωm−σ +1 f defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [0, 1]. Let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < ασ −2 < σ − 2. αj f stands for the abstract left Caputo fractional derivative of f of order Here k j D∗−1 α j anchored at −1. We consider the abstract left fractional linear differential operator v 

αj  , (3.41) α j (x) k j D∗−1 L ∗ := j=h

and suppose, throughout [0, 1],

Set

L ∗ ( f ) ≥ 0.

(3.42)

2 λ j :=

k j (z) dz, j = 1, ..., σ − 2,

(3.43)

0

and λ0 = 1; along with   s j := sup αh−1 (x) α j (x) , j = h, ..., u.

(3.44)

−1≤x≤1

Then, for any n ∈ N there exists a spline Q n (x) ∈ Sm (n ([−1, 1])) such that L ∗ (Q n ) ≥ 0, throughout [0, 1] ,

(3.45)

and (i)      αj αj f (x) − k j D∗−1 Q n (x) ≤ λ j max  k j D∗−1

−1≤x≤1

C2∗ n σ − j−1

 ωm−σ +1

when j = h + 1, ..., σ − 2; (ii) if h = 0 and j = 1, ..., h, we have      αj αj f (x) − k j D∗−1 Q n (x) ≤ max  k j D∗−1

−1≤x≤1

f

(σ −1)

 1 , , n (3.46)

3.3 Main Results

59

⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣

v  j∗ =h

 s j∗ λ j∗

⎤ ⎥ ⎥   αj xh  + λj⎥ max  k j D∗−1 ⎥; −1≤x≤1 ⎦

h!

(3.47)

(iii) furthermore ⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ max | f (x) − Q n (x)| ≤ ⎢ −1≤x≤1 n σ −1−v ⎣



v 

sjλj

⎤ ⎥ ⎥ + 1⎥ ⎥ . (3.48) ⎦

j=h

h!

Proof Let Q n be as in Lemma 3.5. Let also α j > 0, j =1, ...,  σ − 2, such that α0 = 0 < α1 < 1 < α2 < 2 < ... < ασ −2 < σ − 2. That is α j = j, j = 0, 1, ..., σ − 2. αj αj f and k j D∗−1 Qn , We consider the abstract left Caputo fractional derivatives k j D∗−1 j = 1, ..., σ − 2. We notice that (x ∈ [−1, 1]) k α j  k j α j    jD D∗−1 Q n (x) = ∗−1 f (x) −  x    x    k j (x − t) f ( j) (t) dt − k j (x − t) Q ( j) (t) dt  = n     −1

−1

 x       ( j)   k j (x − t) f (t) − Q ( j) (t) dt  ≤ n     −1

x

  (3.9) k j (x − t)  f ( j) (t) − Q (nj) (t) dt ≤

−1

⎛ x ⎞ ⎝ k j (x − t) dt ⎠ −1

⎛ x+1 ⎞ ⎝ k j (z) dz ⎠ 0

C2∗ n σ − j−1 C2∗

n

 ωm−σ +1

ω σ − j−1 m−σ +1



f

(σ −1)

f (σ −1) ,

1 , n

1 n

 =

 ≤

(3.49)

60

3 Spline Abstract Fractional Monotone Approximation

⎛ ⎝

2

⎞ k j (z) dz ⎠



C2∗

n

ω σ − j−1 m−σ +1

 1 . n

(3.50)

 1 , n

(3.51)

f (σ −1) ,

0

We have proved that k α j    k j α j  jD D∗−1 Q n (x) ≤ ∗−1 f (x) − ⎛ 2 ⎞ ⎝ k j (z) dz ⎠



C2∗

n

ω σ − j−1 m−σ +1

f (σ −1) ,

0

∀ x ∈ [−1, 1], j = 1, ..., σ − 2, n ≥ 1. Hence it holds      αj αj max  k j D∗−1 f (x) − k j D∗−1 Q n (x) ≤

(3.52)

−1≤x≤1

λj



C2∗

n

ω σ − j−1 m−σ +1

where

f (σ −1) ,

 1 , j = 1, ..., σ − 2, n ≥ 1, n

2 λ j :=

k j (z) dz, j = 1, ..., σ − 2, as in (3.43.). 0

Inequality (3.52) is valid when j = 0 by (3.9) and (3.20), and we can set λ0 = 1. Set ⎞ ⎛    v 1 ⎝ γn := C2∗ ωm−σ +1 f (σ −1) , s j λ j n j−σ +1 ⎠ . (3.53) n j=h I. Suppose, throughout [0, 1], αh (x) ≥ α > 0. Let Q n (x) ∈ Sm (n ([−1, 1])), x ∈ [−1, 1], as in Lemma 3.5, so that      (3.52.) αj  αj Q n (x) ≤ f (x) + γn (h!)−1 x h − k j D∗−1 max k j D∗−1

−1≤x≤1

λj

C2∗

n

ω σ − j−1 m−σ +1



f (σ −1) ,

(3.54)

 1 , j = 0, 1, ..., σ − 2. n

When j = h + 1, ..., σ − 2, we get      αj αj max  k j D∗−1 f (x) − k j D∗−1 Q n (x) ≤ λ j

−1≤x≤1

C2∗ n σ − j−1

 ωm−σ +1

f

(σ −1)

 1 , , n (3.55)

3.3 Main Results

61

proving (3.46). For j = 1, ..., h = 0, we have that    (3.54.)   αj αj f (x) − k j D∗−1 Q n (x) ≤ max  k j D∗−1

−1≤x≤1

  γn C∗ αj x h  + λ j σ −2j−1 ωm−σ +1 max  k j D∗−1 h! −1≤x≤1 n C2∗ ωm−σ +1 h!



C2∗ ωm−σ +1

f (σ −1) ,

1 n

 =

⎞ ⎛   v   1 αj ⎝ f (σ −1) , s j∗ λ j∗ n j∗ −σ +1 ⎠ max  k j D∗−1 xh  −1≤x≤1 n j =h ∗

+ λj 



1 f (σ −1) , n





C2∗ n σ − j−1

ωm−σ +1

f

(σ −1)

1 , n

 =

(3.56)

⎞ ⎤ ⎛ v    λ α 1 j   j ⎣ ⎝ s j∗ λ j∗ n j∗ −σ +1 ⎠ max  k j D∗−1 x h  + σ − j−1 ⎦ ≤ h! −1≤x≤1 n ⎡

j∗ =h

⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣

v  j∗ =h

 s j∗ λ j∗ h!

⎤ ⎥ ⎥   αj max  k j D∗−1 xh  + λj⎥ ⎥, −1≤x≤1 ⎦

proving (3.47). When j = 0 from (3.54) we get   C2∗ max  f (x) + γn (h!)−1 x h − Q n (x) ≤ σ −1 ωm−σ +1 −1≤x≤1 n



f (σ −1) ,

 1 . n

(3.57)

That is γn C2∗ + σ −1 max | f (x) − Q n (x)| ≤ ωm−σ +1 −1≤x≤1 h! n

 f

(σ −1)

1 , n

 =

(3.58)

⎞  ⎛ v    C2∗ ωm−σ +1 f (σ −1) , n1 1 C2∗ ⎝ ≤ s j λ j n j−σ +1 ⎠ + σ −1 ωm−σ +1 f (σ −1) , h! n n j=h C2∗ ωm−σ +1



f (σ −1) ,



⎡

1 1 ⎣ n h!n σ −1−v

⎛ ⎞ v  ⎝ sjλj⎠ + j=h

⎤ 1 ⎦ ≤ n σ −1

(3.59)

62

3 Spline Abstract Fractional Monotone Approximation

⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣



v 

sjλj

j=h

h!

⎤ ⎥ ⎥ + 1⎥ ⎥, ⎦

proving (3.48). Also if 0 ≤ x ≤ 1, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) + v 

αh−1 (x) α j (x)

'

kj

α

j D∗−1 Q n (x) −

kj

γn h!

α

kh

j D∗−1 f (x) −

j=h

αh D∗−1 xh+

γn h!

kj

α

j D∗−1 xh

(3.60) ( (3.54.) ≥

(by L ∗ f ≥ 0) γn h!

kh

αh D∗−1 x h − C2∗ ωm−σ +1

γn h!

kh



⎞ ⎛   v 1 ⎝ s j λ j n j−σ +1 ⎠ = f (σ −1) , n j=h

αh D∗−1 x h − γn =: ξ

(if h = 0, then αh = 0, and ξ = 0). If h = 0, then ξ = γn

 kh

⎛ x ⎞  αh D∗−1 xh − 1 = γn ⎝ kh (x − t) dt − 1⎠ = h!

(3.61)

−1

⎛ x+1 ⎛ 1 ⎞ ⎞ γn ⎝ kh (z) dz − 1⎠ ≥ γn ⎝ kh (z) dz − 1⎠ ≥ 0, 0

0

1 by the assumption (3.21): 0 kh (z) dz ≥ 1, when h = 0. Hence in both cases we get L ∗ (Q n (x)) ≥ 0, x ∈ [0, 1] .

(3.62)

II. Suppose on [0, 1] that αh (x) ≤ β < 0. Let Q n (x) ∈ Sm (n ([−1, 1])), x ∈ [−1, 1], as in Lemma 3.5, such that      (3.52.) αj  αj f (x) − γn (h!)−1 x h − k j D∗−1 Q n (x) ≤ max k j D∗−1

−1≤x≤1

(3.63)

3.3 Main Results

63



C2∗

λj

n

ω σ − j−1 m−σ +1

f (σ −1) ,

 1 , j = 0, 1, ..., σ − 2, n ≥ 1. n

Similarly, we obtain again the inequalities of convergence, see (3.46), (3.47) and (3.48). Also if 0 ≤ x ≤ 1, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) − v 

αh−1 (x) α j (x)

'

kj

α

j D∗−1 Q n (x) −

kj

γn h!

α

j D∗−1 f (x) +

j=h

kh

γn h!

αh D∗−1 xh+

kj

α

j D∗−1 xh

( (3.63.) ≤

(by L ∗ f ≥ 0) γn − h!

k h αh D∗−1 x h

+ C2∗ ωm−σ +1

−

γn h!

kh



⎞ ⎛   v 1 ⎝ f (σ −1) , s j λ j n j−σ +1 ⎠ = n j=h

αh D∗−1 x h + γn =: θ

(3.64)

(if h = 0, then αh = 0, and θ = 0). If h = 0, then  θ = γn 1 −

kh

αh D∗−1 xh h!



⎡ = γn ⎣1 −

x

⎤ kh (x − t) dt ⎦ =

−1

⎡

⎡ ⎤ ⎤ x+1 1 γn ⎣1 − kh (z) dz ⎦ ≤ γn ⎣1 − kh (z) dz ⎦ ≤ 0. 0

0

Hence again in both cases it holds L ∗ (Q n (x)) ≥ 0, ∀ x ∈ [0, 1] .  

(3.65) 

We also give Theorem 3.13 Let h, v, σ, m be integers, 2 ≤ σ ≤ m, n ∈ N, with h even, 0 ≤ h ≤ v ≤ σ −2 and let f ∈ C σ −1 ([−1, 1]) , with f (σ −1) having modulus of smoothness ωm−σ +1 f (σ −1) , δ there, δ > 0. Let α j (x), j = h, h + 1, ..., v be real functions, defined and bounded on [−1, 1] and suppose αh (x) is either ≥ α > 0 or ≤ β < 0 on [−1, 0]. Let the real numbers α0 = 0 < α1 < 1 < α2 < 2 < ... < ασ −2 < σ − 2.

64

3 Spline Abstract Fractional Monotone Approximation α

Here k j D1−j f stands for the abstract right Caputo fractional derivative of f of order α j anchored at 1. We consider the abstract right fractional linear differential operator L ∗ :=

v 

α j (x)

k j

α  D1−j ,

(3.66)

j=h

and suppose, throughout [−1, 0], L ∗ ( f ) ≥ 0. Set

(3.67)

2 λ j :=

k j (z) dz, j = 1, ..., σ − 2,

(3.68)

0

and λ0 = 1; along with   s j := sup αh−1 (x) α j (x) , j = h, ..., v.

(3.69)

−1≤x≤1

Then, for any n ∈ N there exists a spline Q n (x) ∈ Sm (n [−1, 1]) such that L ∗ (Q n ) ≥ 0, throughout [−1, 0] ,

(3.70)

and (i)      α α max  k j D1−j f (x) − k j D1−j Q n (x) ≤ λ j

−1≤x≤1

C2∗

n

ω σ − j−1 m−σ +1



f (σ −1) ,

 1 , n (3.71)

when j = h + 1, ..., σ − 2; (ii) if h = 0 and j = 1, ..., h, we have      α α max  k j D1−j f (x) − k j D1−j Q n (x) ≤

−1≤x≤1

⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣

v  j∗ =h

 s j∗ λ j∗ h!

⎤ ⎥ ⎥   α max  k j D1−j x h  + λ j ⎥ ⎥; −1≤x≤1 ⎦

(3.72)

3.3 Main Results

65

(iii) furthermore ⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ max | f (x) − Q n (x)| ≤ ⎢ −1≤x≤1 n σ −1−v ⎣



v 

sjλj

j=h

h!

⎤ ⎥ ⎥ + 1⎥ ⎥ . (3.73) ⎦

Proof Let Q n be as in Lemma 3.5. Let also α j > 0, j =1, ...,  σ − 2, such that α0 = 0 < α1 < 1 < α2 < 2 < ... < ασ −2 < σ − 2. That is α j = j, j = 0, 1, ..., σ − 2. α α We consider the abstract right Caputo fractional derivatives k j D1−j f and k j D1−j Q n , j = 1, ..., σ − 2. We notice that (x ∈ [−1, 1]) k α j      j D f (x) − k j D α j Q n (x) = 1− 1−     1 1   (−1) j k j (t − x) f ( j) (t) dt − (−1) j k j (t − x) Q ( j) (t) dt  = n     x

(3.74)

x

 1       ( j)   k j (t − x) f (t) − Q ( j) (t) dt  ≤ n     x

1

  (3.9) k j (t − x)  f ( j) (t) − Q (nj) (t) dt ≤

x

⎛ 1 ⎞ ⎝ k j (t − x)⎠



C2∗

n

f (σ −1) ,

ω σ − j−1 m−σ +1

1 n

 =

x

⎞ ⎛ 1−x ⎝ k j (z) dz ⎠

C2∗ n σ − j−1

 ωm−σ +1

f

(σ −1)

1 , n

 ≤

0

⎛ ⎝

2 0

We have proved that

⎞ k j (z) dz ⎠

C2∗

n

ω σ − j−1 m−σ +1



f (σ −1) ,

 1 . n

(3.75)

66

3 Spline Abstract Fractional Monotone Approximation

k α j      j D f (x) − k j D α j Q n (x) ≤ 1− 1− ⎛ 2 ⎞ ⎝ k j (z) dz ⎠



C2∗ n σ − j−1

ωm−σ +1

f

(σ −1)

 1 , , n

(3.76)

0

∀ x ∈ [−1, 1], j = 1, ..., σ − 2, n ≥ 1. Hence it holds      α α max  k j D1−j f (x) − k j D1−j Q n (x) ≤

(3.77)

−1≤x≤1

λj



C2∗

n

ω σ − j−1 m−σ +1

where

f (σ −1) ,

 1 , j = 1, ..., σ − 2, n ≥ 1, n

2 λ j :=

k j (z) dz, j = 1, ..., σ − 2, as in (3.68.). 0

Inequality (3.77) is valid when j = 0 by (3.9) and (3.20), and we can set λ0 = 1. Set ⎞ ⎛    v 1 ⎝ γn := C2∗ ωm−σ +1 f (σ −1) , s j λ j n j−σ +1 ⎠ . (3.78) n j=h I. Suppose, throughout [−1, 0], αh (x) ≥ α > 0. Let Q n (x) ∈ Sm (n ([−1, 1])), x ∈ [−1, 1], as in Lemma 3.5, so that      (3.77.) α  α max k j D1−j f (x) + γn (h!)−1 x h − k j D1−j Q n (x) ≤

−1≤x≤1

λj

C2∗

n

ω σ − j−1 m−σ +1



f (σ −1) ,

(3.79)

 1 , j = 0, 1, ..., σ − 2. n

When j = h + 1, ..., σ − 2, we get      α α max  k j D1−j f (x) − k j D1−j Q n (x) ≤ λ j

−1≤x≤1

C2∗

n

ω σ − j−1 m−σ +1

proving (3.71). For j = 1, ..., h = 0, we have that    (3.79.)   α α max  k j D1−j f (x) − k j D1−j Q n (x) ≤

−1≤x≤1



f (σ −1) ,

 1 , n (3.80)

3.3 Main Results

67

  γn C∗ α max  k j D1−j x h  + λ j σ −2j−1 ωm−σ +1 h! −1≤x≤1 n C2∗ ωm−σ +1 h!



C2∗ ωm−σ +1

1 n

f (σ −1) ,

 =

⎞ ⎛   v   1 α ⎝ s j∗ λ j∗ n j∗ −σ +1 ⎠ max  k j D1−j x h  + f (σ −1) , −1≤x≤1 n j =h ∗

λj 



1 f (σ −1) , n





C2∗ n σ − j−1

ωm−σ +1

f

(σ −1)

1 , n

 =

(3.81)

⎡

⎛ ⎞ ⎤ v    λ 1 α j   ⎣ ⎝ s j∗ λ j∗ n j∗ −σ +1 ⎠ max  k j D1−j x h  + σ − j−1 ⎦ ≤ −1≤x≤1 h! n j =h ∗

⎡  ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣

v  j∗ =h

 s j∗ λ j∗ h!

⎤ ⎥ ⎥ k α j h  j   max D1− x + λ j ⎥ ⎥, −1≤x≤1 ⎦

proving (3.72). When j = 0 from (3.79) we get   C2∗ max  f (x) + γn (h!)−1 x h − Q n (x) ≤ σ −1 ωm−σ +1 −1≤x≤1 n

 f

(σ −1)

 1 . , n

(3.82)

That is γn C2∗ + σ −1 max | f (x) − Q n (x)| ≤ ωm−σ +1 −1≤x≤1 h! n

 f

(σ −1)

1 , n

 =

⎞  ⎛ v    C2∗ ωm−σ +1 f (σ −1) , n1 C2∗ j−σ +1 ⎠ (σ −1) 1 ⎝ ≤ sjλjn , + σ −1 ωm−σ +1 f h! n n j=h C2∗ ωm−σ +1



f (σ −1) ,



⎡

1 1 ⎣ σ n h!n −1−v ⎡

 ⎢ C2∗ ωm−σ +1 f (σ −1) , n1 ⎢ ⎢ ⎢ n σ −1−v ⎣

⎛ ⎞ v  ⎝ sjλj⎠ + j=h



v 

sjλj

j=h

h!

⎤

(3.83)

1 ⎦ ≤

n σ −1

⎤ ⎥ ⎥ + 1⎥ ⎥, ⎦

(3.84)

68

3 Spline Abstract Fractional Monotone Approximation

proving (3.73). Also if −1 ≤ x ≤ 0, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) + v 

αh−1 (x) α j (x)

'

kj

α

D1−j Q n (x) −

kj

α

D1−j f (x) −

j=h

γn h! γn h!

kh

αh h D1− x +

kj

α

D1−j x h

(3.85)

( (3.79.) ≥

(by L ∗ f ≥ 0) γn h!

kh

αh h D1− x − C2∗ ωm−σ +1

γn h!

kh



⎞ ⎛   v 1 ⎝ f (σ −1) , s j λ j n j−σ +1 ⎠ = n j=h

αh h D1− x − γn =: ξ1

(if h = 0, then αh = 0, and ξ1 = 0). If h = 0, then ξ1 = γn

 kh

⎛ ⎞  1 αh h D1− x (h is even) − 1 = γn ⎝(−1)h kh (t − x) dt − 1⎠ = h!

(3.86)

x

⎛ 1−x ⎛ 1 ⎞ ⎞ γn ⎝ kh (z) dz − 1⎠ ≥ γn ⎝ kh (z) dz − 1⎠ ≥ 0, 0

by the assumption (3.21):

0

1

kh (z) dz ≥ 1, when h = 0.

0

Hence in both cases we get L ∗ (Q n (x)) ≥ 0, x ∈ [−1, 0] .

(3.87)

II. Suppose on [−1, 0] that αh (x) ≤ β < 0. Let Q n (x) ∈ Sm (n ([−1, 1])), x ∈ [−1, 1], as in Lemma 3.5, so that      (3.77.) α  α max k j D1−j f (x) − γn (h!)−1 x h − k j D1−j Q n (x) ≤

−1≤x≤1

λj

C2∗

n

ω σ − j−1 m−σ +1



f (σ −1) ,

 1 , j = 0, 1, ..., σ − 2, n ≥ 1. n

(3.88)

3.3 Main Results

69

Similarly, we obtain again the inequalities of convergence, see (3.71), (3.72) and (3.73). Also if −1 ≤ x ≤ 0, then αh−1 (x) L ∗ (Q n (x)) = αh−1 (x) L ∗ ( f (x)) − v 

αh−1 (x) α j (x)

'

kj

α

D1−j Q n (x) −

kj

α

D1−j f (x) +

j=h

γn h! γn h!

kh

αh h D1− x +

kj

α

D1−j x h

( (3.88.) ≤

(by L ∗ f ≥ 0) −

γn h!

kh

αh h D1− x + C2∗ ωm−σ +1

−

γn h!

kh



⎞ ⎛   v 1 ⎝ f (σ −1) , s j λ j n j−σ +1 ⎠ = n j=h

αh h D1− x + γn =: θ1

(3.89)

(if h = 0, then αh = 0, and θ1 = 0). If h = 0, then  θ1 = γn 1 −

kh

αh h D1− x h!



⎡ = γn ⎣1 −

1

⎤ kh (t − x) dt ⎦ =

x

⎡ γn ⎣1 −

1−x

⎤

⎡

kh (z) dz ⎦ ≤ γn ⎣1 −

0

1

⎤ kh (z) dz ⎦ ≤ 0.

0

Hence again in both cases it holds L ∗ (Q n (x)) ≥ 0, ∀ x ∈ [−1, 0] .

(3.90)

Comment 3.14 Clearly Theorem 3.12 generalizes Theorem 3.7, and Theorem 3.13 generalizes Theorem 6.2, p. 57 of [2]. Furthermore there, the approximating spline Q n depends on f, γn , h; which γn depends on n, C2∗ , v, s j , λ j , σ ; which λ j depends on k j . I.e. polynomial spline Q n among others depends on the type of fractional calculus we use. Consequently, Theorem 3.12 is valid for the following left fractional linear differential operators: (1) v ' (  γ (3.91) α j (x) C Dρ,α j ,ω,−1+ , L ∗1 := j=h

70

3 Spline Abstract Fractional Monotone Approximation

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (3.26)); (2) v 

αj  , (3.92) L ∗2 := α j (x) D−1∗ j=h

(see (3.33)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and (3) v 

αj  , (3.93) α j (x) Cω F D−1+ L ∗3 := j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Similarly, Theorem 3.13 is valid for the following right fractional linear differential operators: (1) ∗ v ( '  C γ (3.94) L ∗∗ := α D (x) j 1 ρ,α j ,ω,1− , j=h

where ρ > 0, γ < 0, and ω > 0 large enough (from Prabhakar fractional calculus, see (3.27)); (2) ∗ v 

α  L ∗∗ := α j (x) D1−j , (3.95) 2 j=h

(see (3.34)) where γ j > 0, j = 1, ..., m; λ = 1; and small enough ω j < 0, j = 1, ..., m (from generalized non-singular fractional calculus); and (3) ∗ v 

α  (3.96) L ∗∗ α j (x) Cω F D1−j , 3 := j=h

with ω < 0, sufficiently small (from parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Our developed abstract simultaneous fractional spline monotone approximation theory with its applications involves weaker conditions than the one with ordinary derivatives ([1]), and can cover many diverse general cases in a multitude of complex settings and environments.

References

71

References 1. Anastassiou, G.A.: Spline monotone approximation with linear differential operators. Approx. Theory Appl. 5(4), 61–67 (1989) 2. Anastassiou, G.A.: Frontiers in Approximation Theory. World Scientific Publ. Corp, New Jersey (2015) 3. Anastassiou, G.A.: Foundations of Generalized Prabhakar-Hilfer fractional Calculus with Applications. Cubo, accepted (2021) 4. Anastassiou, G.A.: Multiparameter Fractional Differentiation with Non Singular Kernel. Issues of Analysis, accepted (2021) 5. Anastassiou, G.A.: Simultaneous High Order Abstract Fractional Polynomial Spline Monotone Approximation and Applications. Submitted (2021) 6. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) 7. De Vore, R., Lorentz, G.: Constructive Approximation. Springer, Heidelberg, New York (1993) 8. Diethelm, K.: The Analysis of Fractional Differential Equations. Lecture Notes in Mathematics, Vol. 2004, 1st edn. Springer, New York, Heidelberg (2010) 9. Giusti, A., et al.: A practical Guide to Prabhakar Fractional Calculus. Fract. Calculus Appl. Anal. 23(1), 9–54 (2020) 10. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler functions, Related Topics and Applications. Springer, Heidelberg (2014) 11. Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1965) 12. Losada, J., Nieto, J.J.: Properties of a New Fractional Derivative without Singular Kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015) 13. Polito, F., Tomovski, Z.: Some properties of Prabhakar-type fractional calculus operators. Fractional Differ. Calculus 6(1), 73–94 (2016) 14. Saxena, R.K., Kalla, S.L.: Ravi Saxena. Multivariate analogue of generalized Mittag-Leffler function. Integr. Transforms Special Funct. 22(7), 533–548 (2011) 15. Schumaker, I.L.: Spline Functions: Basic Theory. Wiley, New York (1981) 16. Srivastava, H.M., Daoust, M.C.: A note on the convergence of Kompe’ de Feriet’s double hypergeometric series. Math. Nachr. 53, 151–159 (1972)

Chapter 4

Abstract Bivariate Left Fractional Monotone Constrained Approximation by Pseudo-polynomials

Here we extend our earlier bivariate high order simultaneous fractional monotone constrained approximation theory by pseudo-polynomials to abstract bivariate high order simultaneous fractional monotone constrained approximation by pseudopolynomials, with applications to bivariate Prabhakar fractional calculus and nonsingular kernel fractional calculi. We cover the left side of this constrained approximation. So we deal with the following general two-dimensional problem: Let f be a two variable continuously differentiable real valued function of a given order, let L ∗ be a linear left abstract fractional mixed partial differential operator and suppose that L ∗ ( f ) ≥ 0 on a critical region. Then for specific and sufficiently large n, m ∈ N, we can find a sequence of pseudo-polynomials Q ∗n,m in two variables with the   property L ∗ Q ∗n,m ≥ 0 on this critical region such that f is approximated with rates fractionally and simultaneously by Q ∗n,m in the uniform norm on the whole domain of f . This constrained approximation is given via inequalities involving the mixed modulus of smoothness ωs,q , s, q ∈ N, of highest order integer partial derivative of f . It follows [7].

4.1 Introduction The topic of monotone approximation theory started in [21] and it has become a major trend of approximation theory. A typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥ 0 by polynomials having this property. In [8] the authors replaced the kth derivative with a linear differential operator of order k. We mention this motivating result. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_4

73

74

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

Theorem 4.1 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p and  let f be  a real function, f ( p) continuous in [−1, 1] with modulus of continuity ω1 f ( p) , x there. Let a j (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume ah (x) is either ≥ some number α > 0 or ≤ some number β < 0 throughout [−1, 1]. Consider the operator  j  k  d a j (x) L= (4.1) d xj j=h and suppose, throughout [−1, 1], L ( f ) ≥ 0.

(4.2)

Then, for every integer n ≥ 1, there is a real polynomial Q n (x) of degree ≤ n such that (4.3) L (Q n ) ≥ 0 throughout [−1, 1] 

and max | f (x) − Q n (x)| ≤ Cn

−1≤x≤1

k− p

ω1

f

( p)

 1 , , n

(4.4)

where C is independent of n or f . Next let n, m ∈ Z+ , Pθ denote the space of algebraic polynomials of degree ≤ θ . Consider the tensor product spaces Pn ⊗ C ([−1, 1]) , C ([−1, 1]) ⊗ Pm and their sum Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm , that is Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm = ⎧ n ⎨ ⎩

i=0

x i Ai (y) +

m  j=0

⎫ ⎬

B j (x) y j ; Ai , B j ∈ C ([−1, 1]) , x, y ∈ [−1, 1] . (4.5) ⎭

This is the space of pseudo-polynomials of degree ≤ (n, m), first introduced by A. k+l Marchaud in 1924–1927 (see [16, 17]). Here f (k,l) denotes ∂∂x k ∂ yf l , the (k, l)-partial derivative of f .   Here we consider the space C r, p [−1, 1]2 = { f : [−1, 1]2 → R; f (k,l) is con  tinuous for 0 ≤ k ≤ r , 0 ≤ l ≤ p}. Let f ∈ C [−1, 1]2 ; for δ1 , δ2 ≥ 0, define the mixed modulus of smoothness of order (s, q), s, q ∈ N (see [20, pp. 516–517]) by   q ωs,q ( f ; δ1 , δ2 ) ≡ sup x sh 1 ◦ y h 2 f (x, y) : (x, y) ,  (x + sh 1 , y + qh 2 ) ∈ [−1, 1]2 , |h i | ≤ δi , i = 1, 2 .

(4.6)

4.1 Introduction

75

Here s x h 1

q ◦ y h 2

f (x, y) ≡

q s  

(−1)

σ =0 μ=0

s+q−σ −μ

   s q f (x + σ h 1 , y + μh 2 ) σ μ (4.7)

is a mixed difference of order (s, q) . We mention   Theorem 4.2 (Gonska [11]). Let r, p ∈ Z+ , s, q ∈ N, and f ∈ C r, p [−1, 1]2 . Let n, m ∈ N with n ≥ max {4 (r + 1) , r + s} and  m ≥ max  {4 ( p + 1) , p + q} . Then there exists a linear operator Q n,m from C r, p [−1, 1]2 into the space of pseudopolynomials (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that   (k,l)   (4.8) (x, y) ≤  f − Q n,m ( f )   Mr,s · M p,q (n (x))r −k · (m (y)) p−l · ωs,q f (r, p) ; n (x) , m (y) , for all (0, 0) ≤ (k, l) ≤ (r, p), x, y ∈ [−1, 1], where √ 1 1 − z2 + 2 , θ = n, m; z = x, y ∈ [−1, 1] . θ (z) = θ θ

(4.9)

The constants Mr,s , M p,q , are independent of f , (x, y) and (n, m); they depend only on (r, s), ( p, q), respectively. (r, p)

See also [12], saying that Q n,m ( f ) is continuous on [−1, 1]2 . We need the following result which is an easy consequence of the last theorem (see [20, p. 517]).   Corollary 4.3 Let r, p ∈ Z+ , s, q ∈ N, and f ∈ C r, p [−1, 1]2 . Let n, m ∈ N with n ≥ max {4 (r + 1) , r + s} and m ≥ max {4 ( p + 1) , p + q} . Then there exists a pseudopolynomial Q n,m ≡ Q n,m ( f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that      (k,l) C (r, p) 1 1  ≤ f , , (4.10) f − Q (k,l) · ω ; s,q n,m ∞ n r −k m p−l n m  depends only on r, p, s, q. for all (0, 0) ≤ (k, l) ≤ (r, p). Here the constant C Corollary 4.3 was used in the proof of the motivational result that follows. Theorem 4.4 ([1]) Let h1 , h 2 , v1 ,v2 , r, p be integers, 0 ≤ h 1 ≤ v1 ≤ r , 0 ≤ h 2 ≤ f ∈ Cr, p [−1, 1]2 , with f (r, p) having a mixed modulus of smoothv2 ≤ p and  let (r, p) ; x, y there, s, q ∈ N. Let αi, j (x, y), i = h 1 , h 1 + 1, ..., v1 ; j = ness ωs,q f h 2 , h 2 + 1, ..., v2 be real-valued functions, defined and bounded in [−1, 1]2 and suppose αh 1 h 2 is either ≥ α > 0 or ≤ β < 0 throughout [−1, 1]2 . Take the operator

76

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation … v1  v2 

L=

αi j (x, y)

i=h 1 j=h 2

∂ i+ j ∂xi ∂y j

(4.11)

and assume, throughout [−1, 1]2 that L ( f ) ≥ 0.

(4.12)

Then for any integers n, m with n ≥ max {4 (r + 1) , r + s}, m ≥ max{4 ( p + 1) , p + q}, there exists a pseudopolynomial Q n,m ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm )   such that L Q m,n ≥ 0 throughout [−1, 1]2 and  (k,l)  f − Q (k,l)  n,m

∞

C ≤ r −v p−v · ωs,q n 1m 2

 f

(r, p)

1 1 ; , n m

 ,

(4.13)

for all (0, 0) ≤ (k, l) ≤ (h 1 , h 2 ). Moreover we get   (k,l) f − Q (k,l)  n,m

∞

≤

C · ωs,q r −k n m p−l



1 1 f (r, p) ; , n m

 ,

(4.14)

for all (h 1 + 1, h 2 + 1) ≤ (k, l) ≤ (r, p). Also (4.14) is valid whenever 0 ≤ k ≤ h 1 , h 2 + 1 ≤ l ≤ p or h 1 + 1 ≤ k ≤ r , 0 ≤ l ≤ h 2 . Here C is a constant independent of f and n, m. It depends only on r, p, s, q, L . We are also motivated by Anastassiou [2]. We mention

  Definition 4.5 (see [15]) Let [−1, 1]2 ; α1 , α2 > 0; α = (α1 , α2 ), f ∈ C [−1, 1]2 , x = (x1 , x2 ), t = (t1 , t2 ) ∈ [−1, 1]2 . We define the left mixed Riemann-Liouville fractional two dimensional integral of order α 

α I−1+



1 f (x) :=

(α1 ) (α2 )

x1 x2

(x1 − t1 )α1 −1 (x2 − t2 )α2 −1 f (t1 , t2 ) dt1 dt2 ,

−1 −1

(4.15) with x1 , x2 > −1. α Notice here that I−1+ (| f |) < ∞. Definition 4.6 ([3, pp. 1–17]) Let α1 , α2 > 0 with α1 = m 1 , α2 = m 2 , ( · ceiling of the number). Let here f ∈ C m 1 ,m 2 [−1, 1]2 . We consider the left Caputo type fractional partial derivative: (α1 ,α2 ) f (x) := D∗(−1)

1 ·

(m 1 − α1 ) (m 2 − α2 )

4.1 Introduction

x1 x2

77

(x1 − t1 )m 1 −α1 −1 (x2 − t2 )m 2 −α2 −1

−1 −1

∂ m 1 +m 2 f (t1 , t2 ) dt1 dt2 , ∂t1m 1 ∂t2m 2

(4.16)

∀ x = (x1 , x2 ) ∈ [−1, 1]2 , where is the gamma function ∞

(ν) =

e−t t ν−1 dt, ν > 0.

(4.17)

0

We set

(0,0) f (x) := f (x) , ∀ x ∈ [−1, 1]2 ; D∗(−1) (m 1 ,m 2 ) D∗(−1) f (x) :=

∂ m 1 +m 2 f (x) , ∀ x ∈ [−1, 1]2 . ∂ x1m 1 ∂ x2m 2

(4.18) (4.19)

Definition 4.7 We also set (0,α2 ) D∗(−1)

(α1 ,0) D∗(−1)

1 f (x) :=

(m 2 − α2 )

1 f (x) :=

(m 1 − α1 )

x2

(x2 − t2 )m 2 −α2 −1

∂ m 2 f (x1 , t2 ) dt2 , ∂t2m 2

(4.20)

(x1 − t1 )m 1 −α1 −1

∂ m 1 f (t1 , x2 ) dt1 , ∂t1m 1

(4.21)

(x2 − t2 )m 2 −α2 −1

∂ m 1 +m 2 f (x1 , t2 ) dt2 , ∂ x1m 1 ∂t2m 2

(4.22)

(x1 − t1 )m 1 −α1 −1

∂ m 1 +m 2 f (t1 , x2 ) dt1 . ∂t1m 1 ∂ x2m 2

(4.23)

−1

x1 −1

and (m 1 ,α2 ) D∗(−1)

(α1 ,m 2 ) D∗(−1)

1 f (x) :=

(m 2 − α2 )

1 f (x) :=

(m 1 − α1 )

x2 −1

x1 −1

In [3, pp. 1–17], we extended Theorem 4.4 to the fractional level. Indeed there L is replaced by L ∗ , a linear left Caputo fractional mixed partial differential operator. Now the monotonicity property holds true only on the critical square of [0, 1]2 . Simultaneously fractional convergence remains true on all of [−1, 1]2 . So we have proved there Theorem 4.8 Let h 1, h 2 , v1 , v2 , r, p be integers, 0 ≤ h 1 ≤ v1 ≤ r , 0 ≤ h 2 ≤ v2 ≤ p and let f ∈ C r, p [−1, 1]2 , with f (r, p) having a mixed modulus of smoothness ωs,q f (r, p) ; x, y there, s, q ∈ N. Let αi j (x, y), i = h 1 , h 1 + 1, ..., v1 ; j =

78

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

h 2 , h 2 + 1, ..., v2 be real valued functions, defined and bounded in [−1, 1]2 and suppose αh 1 h 2 is either ≥ α > 0 or ≤ β < 0 throughout [0, 1]2 . Here n, m ∈ N : n ≥ max {4 (r + 1) , r + s}, m ≥ max {4 ( p + 1) , p + q} . Set li j :=

sup (x,y)∈[−1,1]2

  −1  α h 1 h 2 (x, y) αi j (x, y) < ∞,

(4.24)

  for all h 1 ≤ i ≤ v1 , h 2 ≤ j ≤ v2 . Let α1i , α2 j ≥ 0 with α1i = i, α2 j = j, i = 0, 1, ..., r ; j = 0, 1, ..., p, ( · ceiling of the number), α10 = 0, α20 = 0. Consider the left fractional bivariate differential operator ∗

L :=

v1  v2 

(α1i ,α2 j ) αi j (x, y) D∗(−1) .

(4.25)

i=h 1 j=h 2

Assume L ∗ f (x, y) ≥ 0, on [0, 1]2 . Then there exists Q ∗n,m ≡ Q ∗n,m ( f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that L ∗ Q ∗n,m (x, y) ≥ 0, on [0, 1]2 . Furthermore it holds: (1)

   (α1i ,α2 j )  (α1i ,α2 j ) D∗(−1) ( f ) − D∗(−1) Q ∗n,m 

∞,[−1,1]2

≤

   (i+ j)−(α1i +α2 j ) C2 (r, p) 1 1   f , , · ω ; s,q n m

(i − α1i + 1) j − α2 j + 1 n r −i m p− j

(4.26)

 is a constant that depends only on r, p, s, q; (h 1 + 1, h 2 + 1) ≤ (i, j) ≤ where C (r, p), or 0 ≤ i ≤ h 1 , h 2 + 1 ≤ j ≤ p, or h 1 + 1 ≤ i ≤ r , 0 ≤ j ≤ h 2 , (2)

   (α1i ,α2 j )  (α11 ,α2 j )  D∗(−1) ( f ) − D∗(−1) Q ∗n,m 

∞,[−1,1]2

ci j · ωs,q r −v 1 n m p−v2



1 1 f (r, p) ; , n m

≤

 ,

(4.27)

4.1 Introduction

79

 Ai j , with for (1, 1) ≤ (i, j) ≤ (h 1 , h 2 ), where ci j = C Ai j := ⎧⎡ ⎞ ⎤⎛ v1  v2 h 1 −i ⎨  h 1 −α1i −k lτ μ 2(τ +μ)−(α1τ +α2μ ) 2 ⎠· ⎣  ⎦⎝ ⎩ k! (h 1 − α1i − k + 1)

(τ − a1τ + 1) μ − α2μ + 1 τ =h 1 μ=h 2 k=0

h − j 2  λ=0

(3)

h 2 −α2 j −λ

2   λ! h 2 − α2 j − λ + 1

   f − Q∗  n,m ∞,[−1,1]2 ≤

+

!(4.28)

(i+ j)−(α1i +α2 j )

2   ,

(i − α1i + 1) j − α2 j + 1 

c00 · ωs,q r −v n 1 m p−v2

f

(r, p)

1 1 ; , n m

 ,

(4.29)

 A00 , with where c00 := C ⎞ ⎛ v1  v2 (τ +μ)−(α1τ +α2μ )  2 1 ⎝   ⎠ + 1, A00 := lτ μ h 1 !h 2 ! τ =h μ=h

(τ − a1τ + 1) μ − α2μ + 1 1

(4)

2

   (0,α2 j )  (0,α2 j ) D∗(−1) ( f ) − D∗(−1) Q ∗n,m 

∞,[−1,1]2

c0 j · ωs,q r −v n 1 m p−v2

 f

(r, p)

1 1 ; , n m

≤

 ,

(4.30)

 A0 j , j = 1, ..., h 2 , with where c0 j = C ⎡

A0 j

⎛ ⎞ v1  v2 (τ +μ)−(α1τ +α2μ )  1 2 ⎝  ⎠ := ⎣ lτ μ h 1 ! τ =h μ=h

(τ − a1τ + 1) μ − α2μ + 1 1

h − j 2  λ=0

(4.31)

2

2h 2 −α2 j −λ   λ! h 2 − α2 j − λ + 1

" 2 j−α2 j  , + 

j − α2 j + 1

and (5)

   (α1i ,0) (α1i ,0) ∗  Q n,m   D∗(−1) ( f ) − D∗(−1)

∞,[−1,1]2

ci0 · ωs,q r −v n 1 m p−v2

 f

(r, p)

1 1 ; , n m

≤

 ,

(4.32)

80

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

 Ai0 , i = 1, ..., h 1 , with where ci0 = C ⎡

⎞ ⎛ v1  v2 (τ +μ)−(α1τ +α2μ )  1 2 ⎝  ⎠ · Ai0 := ⎣ lτ μ h 2 ! τ =h μ=h

(τ − a1τ + 1) μ − α2μ + 1 1

h −i 1  k=0

(4.33)

2

2h 1 −α1i −k k! (h 1 − α1i − k + 1)

" 2i−α1i + .

(i − α1i + 1)

In this chapter we establish bivariate abstract fractional calculus left high order monotone (constrained) approximation theory by pseudo-polynomials of Caputo type, and then we apply our results to bivariate Prabhakar fractional Calculus, bivariate generalized non-singular fractional calculus, and bivariate parametrized CaputoFabrizio non-singular fractional calculus. Next, we build the related necessary fractional calculi background.

4.2 Bivariate Fractional Calculi Here we need to be very specific in preparation for our main results.

4.2.1 Bivariate Abstract Fractional Calculus Let h1 , h 2 , v1 ,v2 , r, p be integers, 0 ≤ h 1 ≤ v1 ≤ r , 0 ≤ h 2 ≤ v2 ≤ p and let f ∈ / Z+ : C r, p [−1, 1]2 .Here h 1 ≤ i ≤ v1 , h 2 ≤ j ≤ v2 . Let α1i , α2 j ≥ 0, α1i , α2 j ∈ α1i = i, α2 j = j, i = 0, 1, ..., r ; j = 0, 1, ..., p, ( · is the ceiling of number), α10 = 0, α20 = 0. Consider also the integrable functions k1i := K α1i , k2 j := K α2 j : [0, 2] → R+ , i = 0, 1, ..., r ; j = 0, 1, ..., p. We consider the abstract left Caputo type bivariate fractional partial derivative of  orders α1i , α2 j k1i k2 j

(α1i ,α2 j )

D∗(−1)

x1 x2 f (x) :=

k1i (x1 − t1 ) k2 j (x2 − t2 ) −1 −1

∀ x = (x1 , x2 ) ∈ [−1, 1]2 . We set

∂ i+ j f (t1 , t2 ) j

∂t1i ∂t2

dt1 dt2 , (4.34)

4.2 Bivariate Fractional Calculi

81 k10 (0,0) k20 D∗(−1)

k1i k2 j

(i, j)

D∗(−1) f (x) :=

f (x) := f (x) ,

∂ i+ j f (x1 ,x2 ) , j ∂ x1i ∂ x2

(4.35) ∀ x = (x1 , x2 ) ∈ [−1, 1]2 .

We also set x2

(i,α2 j )

k1i k2 j

D∗(−1) f (x) :=

k2 j (x2 − t2 )

∂ i+ j f (x1 , t2 )

k1i k2 j

(α1i , j) D∗(−1)

x1 f (x) :=

k1i (x1 − t1 )

dt2 ,

(4.36)

dt1 ,

(4.37)

dt2 ,

(4.38)

∂ i f (t1 , x2 ) dt1 , ∂t1i

(4.39)

j

∂ x1i ∂t2

−1

∂ i+ j f (t1 , x2 )

−1

j

∂t1i ∂ x2

and in particular we define: x2

k10 (0,α2 j ) k2 j D∗(−1)

f (x) :=

k2 j (x2 − t2 ) −1

(α1i ,0) k1i k20 D∗(−1)

x1 f (x) :=

k1i (x1 − t1 ) −1

∂ j f (x1 , t2 ) j

∂t2

∀ x = (x1 , x2 ) ∈ [−1, 1]2 . We will assume that 1 ki h i (z) dz ≥ 1, when h i = 0,

(4.40)

0

where i = 1, 2.     In [3], we got that 0 < h 1 − α1h 1 + 1 , h 2 − α2h 2 + 1 ≤ 1, where is the gamma function, and there it is ki h i (z) = and (4.40) is fulfilled.

z h i −αi hi −1   , i = 1, 2; ∀ z ∈ [0, 2] ,

h i − αi h i

82

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

4.2.2 About Bivariate Fractional Calculus We consider the Prabhakar function (also known as the three parameter Mittag-Leffler function), (see [13, p. 97]; [10]) γ

E α,β (z) =

∞  k=0

(γ )k zk , k! (αk + β)

(4.41)

where α, β > 0, γ ∈ R, z ∈ R, and (γ )k = γ (γ + 1) ... (γ + k − 1) , it is 1 0 . E α,β (z) = (β) Let a, b, c, d ∈ R, a < b, c < d; f ∈ C N1 ,N2 ([a, b] × [c, d]), ρi , μi > 0; γi , / N; i = 1, 2. ωi ∈ R; Ni = μi , μi ∈ We define the bivariate Prabhakar-Caputo left partial fractional derivative of orders (μ1 , μ2 ) as follows (x = (x1 , x2 ) ∈ [a, b] × [c, d]): #

C

(γ ,γ ) D(ρ11 ,ρ22 ),(μ1 ,μ2 ),(ω1 ,ω2 ),(a+,c+)

$

x1 x2

f (x) = a

(x1 − t1 ) N1 −μ1 −1 (x2 − t2 ) N2 −μ2 −1

c

(4.42) N1 +N2 % & % & f , t ∂ (t ) 1 2 −γ −γ dt1 dt2 , E ρ1 ,N1 1 −μ1 ω1 (x1 − t1 )ρ1 E ρ2 ,N2 2 −μ2 ω2 (x2 − t2 )ρ2 ∂t1N1 ∂t2N2 #

with

C

#

C

$ (γ ,γ ) D(ρ11 ,ρ22 ),(0,0),(ω1 ,ω2 ),(a+,c+) f (x) := f (x) ;

$ (γ ,γ ) D(ρ11 ,ρ22 ),(N1 ,N2 ),(ω1 ,ω2 ),(a+,c+) f (x) :=

(4.43) ∂ N1 +N2 f (x1 ,x2 ) , N N ∂ x1 1 ∂ x2 2

where N1 , N2 ∈ N, etc. For the related univariate theory see [5, 18].

4.2.3 About Generalized Non-singular Fractional Calculus Here we use the multivariate analogue of generalized Mittag-Leffler function, see [19], defined for λ, γ j , ρ j , z j ∈ C, Re ρ j > 0 ( j = 1, ..., m) in terms of a multiple series of the form: E

(γ j ) (γ ,...,γ ) , ..., z m ) = E (ρ11 ,...,ρmm ),λ (z 1 , ..., z m ) = (ρ j ),λ (z 1

4.2 Bivariate Fractional Calculi

83

∞ 

(γ1 )k1 ... (γm )km z 1k1 ...z mkm  , m k1 !...km ! ' k1 ,...,km =0

λ+ kjρj

(4.44)

j=1

  where γ j k j is the Pochhammer symbol. By [22, p. 157], (4.44) converges for   Re ρ j > 0, j = 1, ..., m. (γ ,...,γ m ) % ρ ρ & ωθ1 tθ θ , ..., ωθm tθ θ , denoted In particular we will use θ = 1, 2; E (ρθθ 1,...,ρθθ),λ θ (γθ j ) % ρ ρ & by E (ρθ ),λθ ωθ1 tθ θ , ..., ωθm tθ θ , where 0 < ρθ < 1, tθ ≥ 0, λθ > 0, γθ j ∈ R with       γθ j kθ j := γθ j γθ j + 1 ... γθ j + kθ j − 1 , ωθ j ∈ R − {0}, for j = 1, ..., m.   / N and μθ = Nθ ∈ N; θ = 1, 2. Let f ∈ C N1 ,N2 [−1, 1]2 , 0 < μθ ∈ We define the bivariate Caputo type generalized left partial fractional derivative with non-singular kernel of order (μ1 , μ2 ), as follows: (μ ,μ2 )

1 D−1∗



  A (1 − n 1 + μ1 , 1 − N2 + μ2 ) γ1 j ω1 j C A (μ1 ,μ2 ),(λ1 ,λ2 )   D−1∗ f (x) := γ2 j ω2 j (N1 − μ1 ) (N2 − μ2 )

f (x) := 

 −ωθ1 (1 − Nθ + μθ ) ( γθ j ) E (1−Nθ +μθ ),λθ (xθ − tθ )1−Nθ +μθ N − μ θ θ θ=1

x1 x2 ( 2 −1 −1

, ...,

 N1 +N2 ∂ f (t1 , t2 ) −ωθm (1 − Nθ + μθ ) dt1 dt2 , (xθ − tθ )1−Nθ +μθ N1 Nθ − μθ ∂t1 ∂t2N2

(4.45)

∀ x = (x1 , x2 ) ∈ [−1, 1]2 , where A := A (1 − n 1 + μ1 , 1 − N2 + μ2 ) is a normalizing constant. Without loss of generality we assume that A > 0. N1 +N2 (0,0) (N1 ,N2 ) f = f , D−1∗ f = ∂ N1 Nf2 , when N1 , N2 ∈ N, etc. We set D−1∗ ∂ x1 ∂ x2

For the univariate theory see the related [4, 6, 9].

4.2.4 Bivariate Parametrized Caputo-Fabrizio Type Non-singular Kernel Left Partial Fractional Derivative of Orders (µ1 , µ2 )   Let f ∈ C N1 ,N2 [−1, 1]2 , N  μ1 , μ2 > 0, μθ = Nθ ∈ N; ωθ < 0; θ = 1, 2. It is given by (μ1 ,μ2 ) CF (ω1 ,ω2 ) D−1+

f (x) :=

1 (N1 − μ1 ) (N2 − μ2 )

84

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

 N1 +N2  x1 x2 ( 2  ∂ f (t1 , t2 ) (1 − Nθ + μθ ) ωθ exp − dt1 dt2 , (4.46) (xθ − tθ ) N1 N − μ ∂t1 ∂t2N2 θ θ θ=1

−1 −1

(0,0) (N1 ,N2 ) ∀ x = (x1 , x2 ) ∈ [−1, 1]2 , and C(ωF1 ,ω2 ) D−1+ f = f , C(ωF1 ,ω2 ) D−1+ f =

For the univariate case see [6, 14]. We make

∂ N1 +N2 f N N ∂ x1 1 ∂ x2 2

, etc.

Remark 4.9 Fractional Calculi 4.2.2–4.2.4 are special cases of the abstract fractional calculus 4.2.1. The abstract important condition (4.40) is fulfilled by: in Sect. 4.2.2 for large enough ωθ > 0, θ = 1, 2; in Sect. 4.2.3 for small enough ωθ j < 0, θ = 1, 2; j = 1, ..., m; and in Sect. 4.2.4 for small enough ωθ < 0, θ = 1, 2. For details see [6] and Chap. 1.

4.3 Main Results We present the following bivariate abstract fractional monotone (constrained) approximation result. Theorem 4.10 Let h 1 , h 2 , v1, v2 , r, p be integers, 0 ≤ h 1 ≤ v1 ≤ r, 0 ≤ h 2 ≤ v2 ≤ p and f ∈ C r,p [−1, 1]2 , with f (r, p) having a mixed modulus of smoothness  let (r, p) ; δ1 , δ2 , δ1 , δ2 > 0 there, s, q ∈ N. Let αi j (x, y), i = h 1 , h 1 + 1, ..., v1 ; ωs,q f j = h 2 , h 2 + 1, ..., v2 be real-valued functions, defined and bounded in [−1, 1]2 and suppose αh 1 h 2 is either ≥ α > 0 or ≤ β < 0 throughout [0, 1]2 . Here n, m ∈ N : n ≥ max {4 (r + 1) , r + s}, m ≥ max {4 ( p + 1) , p + q}. Set li j :=

sup (x,y)∈[−1,1]2

 −1  α  h 1 h 2 (x, y) αi j (x, y) < ∞

(4.47)

  for all h 1 ≤ i ≤ v1 , h 2 ≤ j ≤ v2 . Let 0 < α1i , α2 j ∈ / N with α1i = i, α2 j = j, j = 1, ..., r ; j = 1, ..., p ( · is the ceiling of the number) and α10 = α20 = 0. Consider the abstract left fractional bivariate differential operator L ∗ :=

v1  v2 

αi j (x, y)

k1i k2 j

(α1i ,α2 j ) D∗(−1) .

(4.48)

i=h1 j=h 2

Assume L ∗ f (x, y) ≥ 0, on [0, 1]2 .There exists a pseudo-polynomial of degree ≤ (n, m) Q ∗n,m := Q ∗n,m ( f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that L ∗ Q ∗n,m (x, y) ≥ 0, on [0, 1]2 . We set

4.3 Main Results

85

2 λ1i :=

2 k1i (z) dz, λ2 j :=

0

k2 j (z) dz,

(4.49)

0

for i = 1, ..., r ; j = 1, ..., p. Set also λ10 = λ20 = 1. Furthermore it holds: (1) if (h 1 + 1, h 2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h 1 , h 2 + 1 ≤ j ≤ p, or h 1 + 1 ≤ i ≤ r , 0 ≤ j ≤ h 2 , then  k1i (α1i ,α2 j ) k2 j D∗(−1) f −

k1i k2 j

  (α1i ,α2 j ) D∗(−1) Q ∗n,m 

∞,[−1,1]2

≤

   C (r, p) 1 1 , λ1i λ2 j r −i p− j ωs,q f ; , n m n m

(4.50)

(2) if (1, 1) ≤ (i, j) ≤ (h 1 , h 2 ), then    s,q f (r, p) ; 1 , 1 Cω n m D∗(−1) ≤ n r −v1 m p−v2 ⎧⎛ v ⎫ ⎞ v2 1 ' ' ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ i =h j =h li∗ j∗ λ1i∗ λ2 j∗ ⎟  ⎬  ⎜∗ 1 ∗ 2 ⎟ k1i (α1i ,α2 j )  h 1 h 2  , (4.51) + λ λ x y  ⎜ ⎟ k2 j D∗(−1) 1i 2 j ⎪ ⎪ ∞,[−1,1]2 ⎝ ⎠ h 1 !h 2 ! ⎪ ⎪ ⎪ ⎪ ⎩ ⎭  k1i (α1i ,α2 j ) k2 j D∗(−1) f −

k1i k2 j

(α1i ,α2 j )



 Q ∗n,m  ∞,[−1,1]2

(3) it holds ⎡

v1 v2 ' '

   s,q f (r, p) ; 1 , 1 ⎢   Cω n m ⎢ i ∗ =h 1  f − Q∗  ⎢ n,m ∞,[−1,1]2 ≤ ⎣ n r −v1 m p−v2

j∗ =h 2

⎤ li∗ j∗ λ1i∗ λ2 j∗

h 1 !h 2 !

⎥ ⎥ + 1⎥ , ⎦ (4.52)

(4) when i = 0, j = 1, ..., h 2 , we get  k10 (0,α2 j ) k2 j D∗(−1) f − ⎡⎛

   s,q f (r, p) ; 1 , 1 Cω n m ≤ n r −v1 m p−v2 ⎤

⎞ li∗ j∗ λ1i∗ λ2 j∗ ⎟   ⎥ j∗ =h 2 ⎥ ⎟ k10 (0,α2 j ) h 2  + λ2 j ⎥ , ⎟ k2 j D∗(−1) y  ∞,[−1,1] ⎦ ⎠ h 1 !h 2 !

v1 v2 ' '

⎢⎜ i =h ⎢⎜ ∗ 1 ⎢⎜ ⎣⎝



k10 (0,α2 j ) ∗  k2 j D∗(−1) Q n,m  ∞,[−1,1]2

(4.53)

86

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

and (5) the case of j = 0, i = 1, ..., h 1 follows, it holds:  k1i (α1i ,0) k20 D∗(−1) f − ⎡⎛



(α1i ,0) ∗  k1i k20 D∗(−1) Q n,m  ∞,[−1,1]2

⎞

v1 v2 ' '

⎢⎜ i =h ⎢⎜ ∗ 1 ⎢⎜ ⎣⎝

   s,q f (r, p) ; 1 , 1 Cω n m ≤ n r −v1 m p−v2 ⎤

li∗ j∗ λ1i∗ λ2 j∗ ⎟   ⎥ ⎥ ⎟ k1i (α1i ,0) h 1  + λ1i ⎥ . ⎟ k20 D∗(−1) x  ∞,[−1,1] ⎦ ⎠ h 1 !h 2 !

j∗ =h 2

(4.54)

Proof By Corollary 4.3 there exists Q n,m ≡ Q n,m ( f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that

 (i, j)  f − Q (i, j)  n,m

∞

   C (r, p) 1 1 , ≤ r −i p− j · ωs,q f ; , n m n m

(4.55)

   depends only for all (0, 0) ≤ (i, j) ≤ (r, p), while Q n,m ∈ C r, p [−1, 1]2 . Here C on r, p, s, q, where n ≥ max {4 (r + 1) , r + s} and m ≥ max {4 ( p + 1) , p + q}, with r, p ∈ Z+ , s, q ∈ N, f ∈ C r, p [−1, 1]2 . (r, p) Indeed by [12] we have that Q n,m is continuous on [−1, 1]2 . We observe the following (i = 1, ..., r ; j = 1, ..., p)  k1i (α1i ,α2 j ) k2 j D∗(−1) f (x1 , x2 ) −

k1i k2 j

  (α1i ,α2 j ) D∗(−1) Q n,m (x1 , x2 ) =

 x x  1  2  ∂ i+ j f (t1 , t2 )  k − t − t dt1 dt2 − k (x ) (x ) 1i 1 1 2 j 2 2  j ∂t1i ∂t2  −1 −1

x1 x2 k1i (x1 − t1 ) k2 j (x2 − t2 )

∂

−1 −1

i+ j

   Q n,m (t1 , t2 )  dt dt 1 2 = j ∂t1i ∂t2 

(4.56)

 x x    1  2  i+ j i+ j   ∂ f , t Q , t ∂ (t ) (t ) 1 2 n,m 1 2  k1i (x1 − t1 ) k2 j (x2 − t2 ) − dt1 dt2  ≤  j j i i ∂t1 ∂t2 ∂t1 ∂t2   −1 −1

x1 x2 −1 −1

   ∂ i+ j f (t , t ) ∂ i+ j Q (t , t )  1 2 n,m 1 2   k1i (x1 − t1 ) k2 j (x2 − t2 )  −  dt1 dt2 ≤ j  ∂t i ∂t j  ∂t i ∂t 1

2

1

2

4.3 Main Results

87

⎛ x x ⎞  1 2 ⎝ k1i (x1 − t1 ) k2 j (x2 − t2 ) dt1 dt2 ⎠ −1 −1

⎛ ⎝

x1

⎞⎛ k1i (x1 − t1 ) dt1 ⎠ ⎝

−1

x2

⎞ k2 j (x2 − t2 ) dt2 ⎠

−1

⎛ x +1 ⎞ ⎛ x +1 ⎞ 1 2 ⎝ k1i (z) dz ⎠ ⎝ k2 j (z) dz ⎠ 0

   C (r, p) 1 1 ωs,q f ; , = n r −i m p− j n m    C (r, p) 1 1 f = ω ; , s,q n r −i m p− j n m

 C ωs,q n r −i m p− j



1 1 f (r, p) ; , n m

(4.57)

 ≤

0

⎛ ⎝

2

⎞⎛ 2 ⎞  k1i (z) dz ⎠ ⎝ k2 j (z) dz ⎠

0

   C (r, p) 1 1 f . ω ; , s,q n r −i m p− j n m

0

We have proved that  k1i (α1i ,α2 j ) k2 j D∗(−1) f (x1 , x2 ) − ⎛ ⎝

2

k1i k2 j

⎞⎛ 2 ⎞  k1i (z) dz ⎠ ⎝ k2 j (z) dz ⎠

0

  (α1i ,α2 j ) D∗(−1) Q n,m (x1 , x2 ) ≤

(4.58)

   C (r, p) 1 1 f , , ω ; s,q n r −i m p− j n m

0

∀ (x1 , x2 ) ∈ [−1, 1]2 ; i = 1, ..., r ; j = 1, ..., p. So we have proved that there exists Q n,m such that  k1i (α1i ,α2 j ) k2 j D∗(−1) ( f ) − ⎛ ⎝

2

k1i k2 j

  (α1i ,α2 j )  Q n,m  D∗(−1)

⎞⎛ 2 ⎞  k1i (z) dz ⎠ ⎝ k2 j (z) dz ⎠

0

∞,[−1,1]2

≤

(4.59)

   C (r, p) 1 1 , ωs,q f ; , n r −i m p− j n m

0

i = 1, ..., r ; j = 1, ..., p. We call

2

λ1i :=

2 k1i (z) dz, λ2 j :=

0

for i = 1, ..., r ; j = 1, ..., p, as in (4.49). We also set λ10 = λ20 = 1.

k2 j (z) dz, 0

(4.60)

88

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

Thus the following inequality is valid in general:  k1i (α1i ,α2 j ) k2 j D∗(−1) ( f ) − λ1i λ2 j

  (α1i ,α2 j )  Q n,m  D∗(−1)

k1i k2 j

∞,[−1,1]2

≤

(4.61)

   C (r, p) 1 1 f , , ω ; s,q n r −i m p− j n m

for i = 0, 1, ..., r ; j = 0, 1, ..., p. Define  s,q ρn,m := Cω



1 1 f (r, p) ; , n m

⎤ ⎡   v1  v2   ⎣ li j λ1i λ2 j n i−r m j− p ⎦ .

(4.62)

i=h 1 j=h 2

(I) Suppose, throughout [0, 1]2 , αh 1 h 2 (x, y) ≥ α > 0. Let Q ∗n,m (x, y) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ), (x, y) ∈ [−1, 1]2 , as in (4.61), so that    k1i (α1i ,α2 j ) x h1 y h2  D f (x, y) + ρn,m − k2 j ∗(−1) h1! h2!

k1i k2 j

(α1i ,α2 j )

D∗(−1)

Q ∗n,m

  (x, y) 

∞,[−1,1]2

   C (r, p) 1 1 =: Ti j , λ1i λ2 j r −i p− j ωs,q f ; , n m n m

≤

(4.63)

for i = 0, 1, ..., r ; j = 0, 1, ..., p. If (h 1 + 1, h 2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h 1 , h 2 + 1 ≤ j ≤ p, or h 1 + 1 ≤ i ≤ r , 0 ≤ j ≤ h 2 we get from the last  k1i (α1i ,α2 j ) k2 j D∗(−1) ( f ) − λ1i λ2 j

k1i k2 j

  (α1i ,α2 j ) D∗(−1) Q ∗n,m 

∞,[−1,1]2

≤

   C (r, p) 1 1 f , , ω ; s,q n r −i m p− j n m

(4.64)

proving (4.50). If (0, 0) ≤ (i, j) ≤ (h 1 , h 2 ), we get that  k1i (α1i ,α2 j ) ρn,m  D k2 j ∗(−1) f + h !h ! 1 2

k1i k2 j

(α1i ,α2 j ) 

D∗(−1)

h1 h2

x y



−

≤ Ti j . That is for (1, 1) ≤ (i, j) ≤ (h 1 , h 2 ), we have

k1i k2 j

(α1i ,α2 j )

D∗(−1)

Q ∗n,m

  (x, y) 

∞,[−1,1]2

(4.65)

4.3 Main Results

89

 k1i (α1i ,α2 j ) k2 j D∗(−1) f −

  (α1i ,α2 j ) D∗(−1) Q ∗n,m 

k1i k2 j

∞,[−1,1]2

≤

 ρn,m  k1i (α1i ,α2 j )  h 1 h 2  + Ti j = x y  k2 j D∗(−1) ∞,[−1,1]2 h 1 !h 2 ! ⎤   ⎡ v1 v2  s,q f (r, p) ; 1 , 1    Cω n m ⎣ li∗ j∗ λ1i∗ λ2 j∗ n i∗ −r m j∗ − p ⎦ h 1 !h 2 ! i =h j =h ∗

  k1i (α1i ,α2 j )  h 1 h 2  x y  k2 j D∗(−1)

 s,q Cω



∞,[−1,1]2

1 1 f (r, p) ; , n m

∗

1

2

   C (r, p) 1 1 = + λ1i λ2 j r −i p− j ωs,q f ; , n m n m

⎧ ⎨

⎡

1 ⎣ ⎩ h 1 !h 2 !

v1  v2 

⎤ li∗ j∗ λ1i∗ λ2 j∗ n i∗ −r m j∗ − p ⎦

i ∗ =h 1 j∗ =h 2

  k1i (α1i ,α2 j )  h 1 h 2  x y  k2 j D∗(−1)

∞,[−1,1]2

λ1i λ2 j + r −i p− j n m

. ≤

  / 'v1 'v2  s,q f (r, p) ; 1 , 1 Cω i ∗ =h 1 j∗ =h 2 li ∗ j∗ λ1i ∗ λ2 j∗ n m r −v p−v 1 2 n m h 1 !h 2 !

(4.66)

.

  k1i (α1i ,α2 j )  h 1 h 2  x y  k2 j D∗(−1)

∞,[−1,1]2

+ λ1i λ2 j ,

proving (4.51). If i = j = 0, from (4.63) we obtain        C (r, p) 1 1  f + ρn,m x h 1 y h 2 − Q ∗  , ≤ r p ωs,q f ; , n,m   h 1 !h 2 ! n m n m ∞,[−1,1]2 and

   C ρn,m  f − Q∗  + r p ωs,q n,m ∞,[−1,1]2 ≤ h 1 !h 2 ! n m

 f

(r, p)

1 1 ; , n m



⎤   ⎡ v1 v2  s,q f (r, p) ; 1 , 1   Cω n m ⎣ li∗ j∗ λ1i∗ λ2 j∗ n i∗ −r m j∗ − p ⎦ h 1 !h 2 ! i =h j =h ∗

+

1

∗

2

   C (r, p) 1 1 f , = ω ; s,q nr m p n m

=

(4.67)

90

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

 s,q Cω



1 1 f (r, p) ; , n m



⎤ v1  v2  1 1 ⎣ li j λ1i λ2 j n i∗ −r m j∗ − p + r p ⎦ ≤ h 1 !h 2 ! i =h j =h ∗ ∗ ∗ ∗ n m ⎡

∗

  s,q f (r, p) ; 1 , 1 Cω n m n r −v1 m p−v2

1

∗

2

 0 'v1

i ∗ =h 1

j∗ =h 2 li ∗ j∗ λ1i ∗ λ2 j∗

h 1 !h 2 !

(4.68)

"

'v2

+1 ,

proving (4.52). Next case is for i = 0, j = 1, ..., h 2 , from (4.63) we get  k10 (0,α2 j ) ρn,m x h 1  D k2 j ∗(−1) f + h !h ! 1 2 λ2 j

k10 (0,α2 j ) h 2 k2 j D∗(−1) y



−

 k10 (0,α2 j ) ∗  k2 j D∗(−1) Q n,m  ∞,[−1,1]2

≤

(4.69)

   C (r, p) 1 1 f , . ω ; s,q n r m p− j n m

Therefore we have that  k10 (0,α2 j ) k2 j D∗(−1) f −



k10 (0,α2 j ) ∗  k2 j D∗(−1) Q n,m  ∞,[−1,1]2

≤

    ρn,m  C k10 (0,α2 j ) h 2  (r, p) 1 1 = y  + λ2 j r p− j ωs,q f ; ,  D ∞,[−1,1]2 h 1 !h 2 ! k2 j ∗(−1) n m n m ⎤   ⎡ v1 v2  s,q f (r, p) ; 1 , 1    Cω n m ⎣ i ∗ −r j∗ − p ⎦ li∗ j∗ λ1i∗ λ2 j∗ n m h 1 !h 2 ! i =h j =h ∗

1

∗

2

   C (r, p) 1 1 = + λ2 j r p− j ωs,q f ; , ∞,[−1,1] n m n m

  k10 (0,α2 j ) h 2  k2 j D∗(−1) y 

 s,q Cω



1 1 f (r, p) ; , n m



⎡ 1'v

1

⎣

i ∗ =h 1

 2 i ∗ −r j∗ − p l λ λ n m i j 1i 2 j ∗ ∗ ∗ ∗ j∗ =h 2

'v2

h 1 !h 2 !

  k10 (0,α2 j ) h 2  k2 j D∗(−1) y 

∞,[−1,1]

+

λ2 j n r m p− j

 ≤

$ #  (r, p) 1 1  ⎡ 'v1 'v2 l λ λ  i j 1i 2 j ∗ ∗ ∗ ∗ Cωs,q f ; n, m i ∗ =h 1 j∗ =h 2 ⎣ n r −v1 m p−v2 h 1 !h 2 !   k10 (0,α2 j ) h 2  k2 j D∗(−1) y 

∞,[−1,1]

 + λ2 j ,

(4.70)

4.3 Main Results

91

proving (4.53). The case of j = 0, i = 1, ..., h 1 , is met similarly as in (4.70). Namely we get  k1i (α1i ,0) k20 D∗(−1) f −



(α1i ,0) ∗  k1i k20 D∗(−1) Q n,m  ∞,[−1,1]2

≤

  0 'v1 'v2  s,q f (r, p) ; 1 , 1 Cω i ∗ =h 1 j∗ =h 2 li ∗ j∗ λ1i ∗ λ2 j∗ n m r −v p−v n 1m 2 h 1 !h 2 !   k1i (α1i ,0) h 1  k20 D∗(−1) x 

∞,[−1,1]

(4.71)

 + λ1i ,

proving (4.54). So if (x, y) ∈ [0, 1]2 , we can write   αh−1 (x, y) L ∗ Q ∗n,m (x, y) = αh−1 (x, y) L ∗ ( f (x, y)) + 1 h2 1 h2 ρn,m h 1 !h 2 !  k1i k2 j

(α1i ,α2 j )

D∗(−1)

k1h 1 k2h 2

v1  v2   (α1h ,α2h )  D∗(−1)1 2 x h 1 y h 2 + αh−1 (x, y) αi j (x, y) · 1 h2 i=h 1 j=h 2

Q ∗n,m

(x, y) −

(by L ∗ f ≥ 0)

(4.63)

≥

⎛ ⎝

k1i k2 j

v1  v2 

(α1i ,α2 j )

D∗(−1)

ρn,m h 1 !h 2 !

k1h 1 k2h 2

ρn,m f (x, y) − h 1 !h 2 !

li j λ1i λ2 j n

m

j− p ⎠

 s,q Cω



i=h 1 j=h 2

⎛k ρn,m ⎝ ⎡# ⎢ ρn,m ⎣

k1h 1 k20

1h 1

k2h 2

 (α1h ,α2h )  D∗(−1)1 2 x h 1 y h 2 h 1 !h 2 !

(α1h ,0) D∗(−1)1 x h 1

(α1i ,α2 j ) 

D∗(−1)

h1 h2

x y

 

(4.72)

 (α1h ,α2h )  D∗(−1)1 2 x h 1 y h 2 − ⎞

i−r

k1i k2 j

$#

k10 k2h 2

1 1 f (r, p) ; , n m

If h 1 = h 2 = 0, then α1h 1 = α2h 2 = 0, and ϕ = 0. If h 1 = 0, and h 2 = 0, then

=

⎞ − 1⎠ =

(0,α2h ) D∗(−1) 2 y h 2

h 1 !h 2 !



$

(4.73) ⎤

⎥ − 1⎦ =: ϕ.

92

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

⎡ ϕ = ρn,m ⎣ ⎡ ρn,m ⎣

y

k10 k2h 2

(0,α2h ) D∗(−1) 2 y h 2 h2!

⎤ − 1⎦ =

⎛ y+1 ⎞  k2h 2 (y − t) dt − 1⎦ = ρn,m ⎝ k2h 2 (z) dz − 1⎠ ≥ ⎤

−1

(4.74)

0

⎛ 1 ⎞  ρn,m ⎝ k2h 2 (z) dz − 1⎠ ≥ 0, 0

by the assumption (4.40). Similarly, we treat the case h 1 = 0, h 2 = 0. When h 1 , h 2 = 0, then we have ⎡⎛ x ⎞⎛ y ⎞ ⎤   ϕ = ρn,m ⎣⎝ k1h 1 (x − t) dt ⎠ ⎝ k2h 2 (y − t) dt ⎠ − 1⎦ = −1

(4.75)

−1

⎡⎛ x+1 ⎤ ⎞ ⎛ y+1 ⎞   ρn,m ⎣⎝ k1h 1 (z) dz ⎠ ⎝ k2h 2 (z) dz ⎠ − 1⎦ ≥ 0

0

⎡⎛ 1 ⎤ ⎞⎛ 1 ⎞   ρn,m ⎣⎝ k1h 1 (z) dz ⎠ ⎝ k2h 2 (z) dz ⎠ − 1⎦ ≥ 0, 0

0

by the assumption (4.40). So in all four cases we get that   L ∗ Q ∗n,m (x, y) ≥ 0, ∀ (x, y) ∈ [0, 1]2 .

(4.76)

(II) Suppose, throughout [0, 1]2 , αh 1 h 2 (x, y) ≤ β < 0. Let Q ∗n,m (x, y) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ), (x, y) ∈ [−1, 1]2 , as in (4.61), so that    k1i (α1i ,α2 j ) x h1 y h2  D f (x, y) − ρn,m − k2 j ∗(−1) h1! h2!

k1i k2 j

(α1i ,α2 j )

D∗(−1)

Q ∗∗ n,m

   C (r, p) 1 1 , λ1i λ2 j r −i p− j ωs,q f ; , n m n m for i = 0, 1, ..., r, j = 0, 1, ..., p.

  (x, y) 

∞,[−1,1]2

≤

(4.77)

4.3 Main Results

93

Similarly, we get as in the first case ≥ α > 0, the inequalities of simultaneous fractional convergence, see {(4.51), (4.66)}, {(4.52), (4.68)}, {(4.53),(4.70)}, {(4.54), (4.71)}. So if (x, y) ∈ [0, 1]2 we can write   αh−1 (x, y) L ∗ Q ∗n,m (x, y) = αh−1 (x, y) L ∗ ( f (x, y)) − 1 h2 1 h2 ρn,m h 1 !h 2 !  k1i k2 j

k1h 1 k2h 2

(α1h1 ,α2h2 ) 

D∗(−1)

h1 h2

x y



+

v1  v2 

αh−1 (x, y) αi j (x, y) · 1 h2

(4.78)

i=h 1 j=h 2

(α1i ,α2 j ) D∗(−1) Q ∗n,m (x, y) −

(by L ∗ f ≥ 0) (4.78)

≤ −

⎛ ⎝

k1i k2 j

ρn,m (α1i ,α2 j ) D∗(−1) f (x, y) + h 1 !h 2 !

ρn,m h 1 !h 2 !

k1h 1 k2h 2

 s,q li j λ1i λ2 j n r −i m p− j ⎠ Cω



i=h 1 j=h 2

⎡ ρn,m ⎣1 − ⎡

#k

⎢ ρn,m ⎣1 −

1h 1

k20

k1h 1 k2h 2

 (α1i ,α2 j )  h 1 h 2  x y D∗(−1)

 (α1h ,α2h )  D∗(−1)1 2 x h 1 y h 2 + ⎞

v1  v2 

k1i k2 j

1 1 f (r, p) ; , n m

 (α1h ,α2h )  D∗(−1)1 2 x h 1 y h 2 h 1 !h 2 !

(α1h ,0) D∗(−1)1 x h 1

$#

k10 k2h 2

=

(4.79)

⎤ ⎦=

(0,α2h ) D∗(−1) 2 y h 2

h 1 !h 2 !



$⎤ ⎥ ⎦ =: ψ.

If h 1 = h 2 = 0, then α1h 1 = α2h 2 = 0, and ψ = 0. If h 1 = 0, and h 2 = 0, then ⎡ ψ = ρn,m ⎣1 −

k10 k2h 2

(0,α2h ) D∗(−1) 2 y h 2 h2!

⎤

⎡

⎦ = ρn,m ⎣1 −

y

⎤ k2h 2 (y − t) dt ⎦ =

−1

⎡

⎡ ⎤ ⎤ y+1 1 k2h 2 (z) dz ⎦ ≤ ρn,m ⎣1 − k2h 2 (z) dz ⎦ ≤ 0, ρn,m ⎣1 − 0

see (4.40). Similarly, we treat the case h 1 = 0, h 2 = 0. When h 1 , h 2 = 0, then we have

0

(4.80)

94

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

⎡

⎛

ψ = ρn,m ⎣1 − ⎝

x

⎞⎛ k1h 1 (x − t) dt ⎠ ⎝

−1

y

⎞⎤ k2h 2 (y − t) dt ⎠⎦ =

−1

⎡

⎛ x+1 ⎞ ⎛ y+1 ⎞⎤   ρn,m ⎣1 − ⎝ k1h 1 (z) dz ⎠ ⎝ k2h 2 (z) dz ⎠⎦ ≤ 0

(4.81)

0

⎡

⎛ 1 ⎞⎛ 1 ⎞⎤   ρn,m ⎣1 − ⎝ k1h 1 (z) dz ⎠ ⎝ k2h 2 (z) dz ⎠⎦ ≤ 0, 0

0

by the assumption (4.40). So in all four cases we proved again that   L ∗ Q ∗n,m (x, y) ≥ 0, ∀ (x, y) ∈ [0, 1]2 .

(4.82)

The proof is complete. Conclusion 4.11 Clearly Theorem 4.10 generalizes Theorem 4.8 to many fractional calculi, opening new avenues of fractional research activity. The approximating pseudo-polynomial Q ∗n,m depends on f, ρn,m , h 1 , h 2 ; which ρn,m depends on  (which depends on r, p, s, q), f, n, m, li j , λ1i , λ2 j ; and which: λ1i depends on k1i C and λ2 j depends on k2 j . That is Q ∗n,m depends on the type of bivariate fractional calculus we use. Consequently, Theorem 4.10 is valid at least for the following important bivariate left fractional linear differential operators: (1) L ∗1 :=

v1  v2  i=h 1 j=h 2

αi j (x, y)

1

C

2 (γ ,γ ) D(ρ1 ,ρ2 ), α ,α ,(ω ,ω ),(−1+,−1+) , 1 2 ( 1i 2 j ) 1 2

(4.83)

where ρ1 , ρ2 > 0, γ1 , γ2 < 0, and ω1 , ω2 > 0 large enough (from bivariate Prabhakar fractional calculus, see (4.42)); (2) v1  v2 1 α ,α 2  ( 1i 2 j ) , (4.84) αi j (x, y) D−1∗ L ∗2 := i=h 1 j=h 2

(see (4.45)) where θ = 1, 2; γθ j > 0, j = 1, ..., m; λθ = 1, 0 < ρθ < 1; and small enough ωθ j < 0, j = 1, ..., m (from bivariate generalized non-singular fractional calculus);

4.3 Main Results

95

and (3) L ∗3 :=

v1  v2 

αi j (x, y)

1

(α1i ,α2 j ) CF (ω1 ,ω2 ) D−1+

2

,

(4.85)

i=h 1 j=h 2

(see (4.46)) for small enough ω1 , ω2 < 0 (from bivariate parametrized CaputoFabrizio non-singular kernel fractional calculus). Our developed bivariate abstract fractional monotone approximation theory by pseudopolynomials with its applications, involves weaker conditions than the one with ordinary partial derivatives, see Theorem 4.4, and can manage many diverse general cases in a multitude of complex settings and environments.

References 1. Anastassiou, G.A.: Monotone Approximation by Pseudopolynomials, Approximation Theory, pp. 5–11. Academic Press, New York (1991) 2. Anastassiou, G.A.: Bivariate Monotone Approximation. Proc. Amer. Math. Soc. 112(4), 959– 964 (1991) 3. Anastassiou, G.A.: Bivariate Left Fractional Pseudo-polynomial Monotone Approximation, Computational Analysis, pp. 1–17. AMAT, Ankara, May 2015, Springer, New York (2016) 4. Anastassiou, G.A.: Multiparameter Fractional Differentiation with Non Singular Kernel. Issues of Analysis, accepted (2021) 5. Anastassiou, G.A.: Foundations of Generalized Prabhakar-Hilfer Fractional Calculus with Applications. Cubo, accepted (2021) 6. Anastassiou, G.A.: Univariate Simultaneous High Order Abstract Fractional Monotone Approximation with Applications. Submitted (2021) 7. Anastassiou, G.A.: Abstract Bivariate Left Fractional Pseudo-Polynomial Monotone Constrained Approximation with Applications. Submitted (2021) 8. Anastassiou, G.A., Shisha, O.: Monotone approximation with linear differential operators. J. Approx. Theory 44, 391–393 (1985) 9. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) 10. Giusti, A., et al.: A practical Guide to Prabhakar Fractional Calculus. Fractional Calculus Appl. Anal. 23(1), 9–54 (2020) 11. Gonska, H.H., Simultaneously approximation by algebraic blending functions, Alfred Haar Memorial Conference, Budapest,: Coloquia Mathematica Soc. Janos Bolyai, 49. North-Holand, Amsterdam, 363–382 (1985) 12. Gonska, H.H.: Personal communication with author, 2-24-2014 13. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler functions. Related Topics and Applications, Springer, Heidelberg, New York (2014) 14. Losada, J., Nieto, J.J.: Properties of a new fractional derivative without singular Kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015) 15. Mamatov, T., Samko, S.: Mixed fractional integration operators in mixed weighted Hölder spaces. Fractional Calculus and Appl. Anal. 13(3), 245–259 (2010) 16. Marchaud, A.: Differences et deerivees d’une fonction de deux variables. C.R. Acad. Sci. 178, 1467–1470 (1924) 17. Marchaud, A.: Sur les derivees et sur les differences des fonctions de variables reelles. J. Math. Pures Appl. 6, 337–425 (1927)

96

4 Abstract Bivariate Left Fractional Monotone Constrained Approximation …

18. Polito, F., Tomovski, Z.: Some properties of Prabhakar-type fractional calculus operators. Fractional Differ. Calculus 6(1), 73–94 (2016) 19. Saxena, R.K., Kalla, S.L.: Ravi Saxena. Multivariate analogue of generalized Mittag-Leffler function. Integr. Transforms Special Funct. 22(7), 533–548 (2011) 20. Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981) 21. Shisha, O.: Monotone approximation. Pac. J. Math. 15, 667–671 (1965) 22. Srivastava, H.M., Daoust, M.C.: A note on the convergence of Kompe’ de Feriet’s double hypergeometric series. Math. Nachr. 53, 151–159 (1972)

Chapter 5

Right Side Abstract Bivariate Monotone Constrained Approximation by Pseudo-polynomials

Here we extend our earlier right side bivariate high order simultaneous fractional monotone constrained approximation theory by pseudo-polynomials to right side abstract bivariate high order simultaneous fractional monotone constrained approximation by pseudo-polynomials, with applications to right side bivariate Prabhakar fractional calculus and non-singular kernel fractional calculi. So we deal with the following general two-dimensional problem: Let f be a two variable continuously differentiable real valued function of a given order, let L∗ be a linear right side abstract fractional mixed partial differential operator and suppose that L∗ (f ) ≥ 0 on the critical region of [−1, 0]2 . Then for specific and sufficiently ∗ in two variables large n, m ∈ N, we can find a sequence of pseudo-polynomials Qn,m  ∗  ∗ with the property L Qn,m ≥ 0 on this critical region of [−1, 0]2 such that f is ∗ in the uniform norm approximated with rates fractionally and simultaneously by Qn,m on the whole domain of f . This constrained approximation is given via inequalities involving the mixed modulus of smoothness ωs,q , s, q ∈ N, of highest order integer partial derivative of f . It follows [7].

5.1 Introduction The topic of monotone approximation theory started in [21] and it has become a major trend of approximation theory. A typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥ 0 by polynomials having this property. In [8] the authors replaced the kth derivative with a linear differential operator of order k. We mention this motivating result.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_5

97

98

5 Right Side Abstract Bivariate Monotone …

Theorem 5.1 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p andlet f be  a real function, f (p) continuous in [−1, 1] with modulus of continuity ω1 f (p) , x there. Let aj (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume ah (x) is either ≥ some number α > 0 or ≤ some number β < 0 throughout [−1, 1]. Consider the operator  j k  d (5.1) aj (x) L= dxj j=h

and suppose, throughout [−1, 1], L (f ) ≥ 0.

(5.2)

Then, for every integer n ≥ 1, there is a real polynomial Qn (x) of degree ≤ n such that (5.3) L (Qn ) ≥ 0 throughout [−1, 1] and max |f (x) − Qn (x)| ≤ Cn

k−p

−1≤x≤1

  (p) 1 , ω1 f , n

(5.4)

where C is independent of n or f . Next let n, m ∈ Z+ , Pθ denote the space of algebraic polynomials of degree ≤ θ . Consider the tensor product spaces Pn ⊗ C ([−1, 1]) , C ([−1, 1]) ⊗ Pm and their sum Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm , that is Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm = ⎧ n ⎨ ⎩

i=0

xi Ai (y) +

m  j=0

⎫ ⎬

Bj (x) yj ; Ai , Bj ∈ C ([−1, 1]) , x, y ∈ [−1, 1] . ⎭

(5.5)

This is the space of pseudo-polynomials of degree ≤ (n, m), first introduced by A. ∂ k+l f Marchaud in 1924-1927 (see [16, 17]). Here f (k,l) denotes ∂x k ∂y l , the (k, l)-partial derivative of f .   Here we consider the space C r,p [−1, 1]2 = {f : [−1, 1]2 → R; f (k,l) is con  tinuous for 0 ≤ k ≤ r, 0 ≤ l ≤ p}. Let f ∈ C [−1, 1]2 ; for δ1 , δ2 ≥ 0, define the mixed modulus of smoothness of order (s, q), s, q ∈ N (see [20, pp. 516–517] ) by   q ωs,q (f ; δ1 , δ2 ) ≡ sup x sh1 ◦y h2 f (x, y) : (x, y) ,  (x + sh1 , y + qh2 ) ∈ [−1, 1]2 , |hi | ≤ δi , i = 1, 2 .

(5.6)

5.1 Introduction

99

Here s x h1

q ◦y h2 f

(x, y) ≡

q s  

(−1)

σ =0 μ=0

s+q−σ −μ

   s q f (x + σ h1 , y + μh2 ) σ μ (5.7)

is a mixed difference of order (s, q) . We mention   Theorem 5.2 (Gonska [11]). Let r, p ∈ Z+ , s, q ∈ N, and f ∈ C r,p [−1, 1]2 . Let n, m ∈ N with n ≥ max {4 (r + 1) , r + s} and  m ≥ max {4 (p + 1) , p + q} . Then there exists a linear operator Qn,m from C r,p [−1, 1]2 into the space of pseudopolynomials (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that   (k,l)   (5.8) (x, y) ≤  f − Qn,m (f )   Mr,s · Mp,q (n (x))r−k · (m (y))p−l · ωs,q f (r,p) ; n (x) , m (y) , for all (0, 0) ≤ (k, l) ≤ (r, p), x, y ∈ [−1, 1], where √ θ (z) =

1 1 − z2 + 2 , θ = n, m; z = x, y ∈ [−1, 1] . θ θ

(5.9)

The constants Mr,s , Mp,q , are independent of f , (x, y) and (n, m); they depend only on (r, s), (p, q), respectively. (r,p)

See also [12], saying that Qn,m (f ) is continuous on [−1, 1]2 . We need the following result which is an easy consequence of the last theorem (see [20, p. 517]).   Corollary 5.3 ([1]) Let r, p ∈ Z+ , s, q ∈ N, and f ∈ C r,p [−1, 1]2 . Let n, m ∈ N with n ≥ max {4 (r + 1) , r + s} and m ≥ max {4 (p + 1) , p + q} . Then there exists a pseudopolynomial Qn,m ≡ Qn,m (f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that      (k,l) C (k,l)  (r,p) 1 1 f , (5.10) − Qn,m ∞ ≤ r−k p−l · ωs,q f ; , n m n m for all (0, 0) ≤ (k, l) ≤ (r, p). Here the constant  C depends only on r, p, s, q. Corollary 5.3 was used in the proof of the motivational result that follows. Theorem 5.4 ([1]) Let h1 , h2 , v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤p and let f  ∈ C r,p [−1, 1]2 , with f (r,p) having a mixed modulus of smoothness ωs,q f (r,p) ; x, y there, s, q ∈ N. Let αi,j (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real-valued functions, defined and bounded in [−1, 1]2 and suppose αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout [−1, 1]2 . Take the operator

100

5 Right Side Abstract Bivariate Monotone … v1  v2 

L=

αij (x, y)

i=h1 j=h2

∂ i+j ∂xi ∂yj

(5.11)

and assume, throughout [−1, 1]2 that L (f ) ≥ 0.

(5.12)

Then for any integers n, m with n ≥ max {4 (r + 1) , r + s}, m ≥ max{4 (p + 1) , p + q}, there exists a pseudopolynomial Qn,m ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm )   such that L Qm,n ≥ 0 throughout [−1, 1]2 and   C (r,p) 1 1 , ≤ r−v p−v · ωs,q f ; , n 1m 2 n m

 (k,l)  f − Q(k,l)  n,m

∞

(5.13)

for all (0, 0) ≤ (k, l) ≤ (h1 , h2 ). Moreover we get   (k,l) f − Q(k,l)  n,m

∞

≤

  C (r,p) 1 1 f , , · ω ; s,q nr−k mp−l n m

(5.14)

for all (h1 + 1, h2 + 1) ≤ (k, l) ≤ (r, p). Also (5.14) is valid whenever 0 ≤ k ≤ h1 , h2 + 1 ≤ l ≤ p or h1 + 1 ≤ k ≤ r, 0 ≤ l ≤ h2 . Here C is a constant independent of f and n, m. It depends only on r, p, s, q, L. We are also motivated by [2]. We mention

  Definition 5.5 (see [14]) Let [−1, 1]2 ; α1 , α2 > 0; α = (α1 , α2 ), f ∈ C [−1, 1]2 , x = (x1 , x2 ), t = (t1 , t2 ) ∈ [−1, 1]2 . We define the right mixed Riemann-Liouville fractional two dimensional integral of order α  α  I1− f (x) :=

1

(α1 ) (α2 )

1 1

(t1 − x1 )α1 −1 (t2 − x2 )α2 −1 f (t1 , t2 ) dt1 dt2 ,

x1 x2

(5.15) with x1 , x2 < 1. α Notice here that I1− (|f |) < ∞. Definition 5.6 ([3, pp. 15–31]) Let α1 , α2 > 0 with  α1 = m1 , α2 = m2 , ( · ceiling of the number). Let here f ∈ C m1 ,m2 [−1, 1]2 . We consider the right Caputo type fractional partial derivative: (α1 ,α2 ) D1− f (x) :=

(−1)m1 +m2 ·

(m1 − α1 ) (m2 − α2 )

5.1 Introduction

1 1

101

(t1 − x1 )m1 −α1 −1 (t2 − x2 )m2 −α2 −1

x1 x2

∂ m1 +m2 f (t1 , t2 ) dt1 dt2 , ∂t1m1 ∂t2m2

(5.16)

∀ x = (x1 , x2 ) ∈ [−1, 1]2 , where is the gamma function ∞

(ν) =

e−t t ν−1 dt, ν > 0.

(5.17)

0

We set

(0,0) f (x) := f (x) , ∀ x ∈ [−1, 1]2 ; D1− (m1 ,m2 ) D1− f (x) := (−1)m1 +m2

∂ m1 +m2 f (x) , ∀ x ∈ [−1, 1]2 . ∂x1m1 ∂x2m2

(5.18) (5.19)

Definition 5.7 ([3], pp. 15-31) We also set (0,α2 ) D1− f

(α1 ,0) D1− f

(−1)m2 (x) :=

(m2 − α2 ) (−1)m1 (x) :=

(m1 − α1 )

1

(t2 − x2 )m2 −α2 −1

∂ m2 f (x1 , t2 ) dt2 , ∂t2m2

(5.20)

(t1 − x1 )m1 −α1 −1

∂ m1 f (t1 , x2 ) dt1 , ∂t1m1

(5.21)

(t2 − x2 )m2 −α2 −1

∂ m1 +m2 f (x1 , t2 ) dt2 , ∂x1m1 ∂t2m2

(5.22)

(t1 − x1 )m1 −α1 −1

∂ m1 +m2 f (t1 , x2 ) dt1 . ∂t1m1 ∂x2m2

(5.23)

x2

1 x1

and (m1 ,α2 ) D1− f

(α1 ,m2 ) D1− f

(−1)m2 (x) :=

(m2 − α2 ) (−1)m1 (x) :=

(m1 − α1 )

1 x2

1 x1

In [3, pp. 15–31], we extended Theorem 5.4 to the fractional level. Indeed there L is replaced by L, a linear right Caputo fractional mixed partial differential operator. Now the monotonicity property holds true only on the critical square of [−1, 0]2 . Simultaneously fractional convergence remains true on all of [−1, 1]2 . So we have proved there Theorem 5.8 Let h1 , h2 , v1, v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ C r,p [−1, 1]2 , with f (r,p) having a mixed modulus of smoothness

102

5 Right Side Abstract Bivariate Monotone …

  ωs,q f (r,p) ; x, y there, s, q ∈ N. Let αij (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real valued functions, defined and bounded in [−1, 1]2 and suppose αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout [−1, 0]2 . Assume that h1 + h2 = 2γ , γ ∈ Z+ . Here n, m ∈ N : n ≥ max {4 (r + 1) , r + s}, m ≥ max {4 (p + 1) , p + q} . Set lij :=

sup (x,y)∈[−1,1]2

 −1  α  h1 h2 (x, y) αij (x, y) < ∞,

(5.24)

  for all h1 ≤ i ≤ v1 , h2 ≤ j ≤ v2 . Let α1i , α2j ≥ 0 with α1i = i, α2j = j, i = 0, 1, ..., r; j = 0, 1, ..., p, ( · ceiling of the number), α10 = 0, α20 = 0. Consider the right fractional bivariate differential operator L :=

v1  v2 

(α1i ,α2j ) αij (x, y) D1− .

(5.25)

i=h1 j=h2

Assume Lf (x, y) ≥ 0, on [−1, 0]2 . Then there exists ∗ ∗ ≡ Qn,m Qn,m (f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) ∗ such that LQn,m (x, y) ≥ 0, on [−1, 0]2 . Furthermore it holds: (1)    (α1i ,α2j ) (α1i ,α2j ) ∗  Qn,m  (f ) − D1− D1−

∞,[−1,1]2

≤

   C2(i+j)−(α1i +α2j ) (r,p) 1 1   f , , · ω ; s,q n m

(i − α1i + 1) j − α2j + 1 nr−i mp−j

(5.26)

where  C is a constant that depends only on r, p, s, q; (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 , (2)    (α1i ,α2j ) (α11 ,α2j ) ∗  Qn,m  ≤ (f ) − D1− D1− 2 ∞,[−1,1]

  cij (r,p) 1 1 f , , · ω ; s,q nr−v1 mp−v2 n m for (1, 1) ≤ (i, j) ≤ (h1 , h2 ), where cij =  CAij , with

(5.27)

5.1 Introduction

103

Aij := ⎧⎡ ⎤  v1  v2 h1 −i ⎨  (τ +μ)−(α1τ +α2μ ) h1 −α1i −k  l 2 2 τ μ ⎣  ⎦ · ⎩ k! (h1 − α1i − k + 1)

(τ − a1τ + 1) μ − α2μ + 1 τ =h μ=h k=0 1

2

h2 −j  λ=0

(3)



h2 −α2j −λ

2   λ! h2 − α2j − λ + 1   f − Q∗  n,m ∞,[−1,1]2 ≤

+

(i+j)−(α1i +α2j )

 (5.28)

2   ,

(i − α1i + 1) j − α2j + 1

  c00 (r,p) 1 1 , · ωs,q f ; , nr−v1 mp−v2 n m

(5.29)

where c00 :=  CA00 , with ⎞ ⎛ v1  v2 (τ +μ)−(α1τ +α2μ )  2 1 ⎝   ⎠ + 1, lτ μ A00 := h1 !h2 !

− a + 1)

μ − α + 1 (τ 1τ 2μ τ =h μ=h 1

(4)

2

   (0,α2j ) (0,α2j ) ∗  (f ) − D1− Qn,m  D1−

∞,[−1,1]2

≤

  c0j (r,p) 1 1 , · ωs,q f ; , nr−v1 mp−v2 n m

(5.30)

where c0j =  CA0j , j = 1, ..., h2 , with ⎡

⎛ ⎞ v1  v2 (τ +μ)−(α1τ +α2μ )  1 2 ⎝  ⎠ A0j := ⎣ lτ μ h1 !

− a + 1)

μ − α + 1 (τ 1τ 2μ τ =h μ=h 1

h2 −j  λ=0

and (5)

(5.31)

2

2h2 −α2j −λ   λ! h2 − α2j − λ + 1



" 2j−α2j  , + 

j − α2j + 1

   (α1i ,0) (α ,0) ∗   D1− (f ) − D1−1i Qn,m

∞,[−1,1]2

  ci0 (r,p) 1 1 f , , · ω ; s,q nr−v1 mp−v2 n m where ci0 =  CAi0 , i = 1, ..., h1 , with

≤ (5.32)

104

5 Right Side Abstract Bivariate Monotone …

⎡

⎞ ⎛ v1  v2 (τ +μ)−(α1τ +α2μ )  1 2 ⎝  ⎠ · Ai0 := ⎣ lτ μ h2 !

− a + 1)

μ − α + 1 (τ 1τ 2μ τ =h μ=h 1

h −i 1  k=0

(5.33)

2

2h1 −α1i −k k! (h1 − α1i − k + 1)



" 2i−α1i + .

(i − α1i + 1)

In this chapter we establish right side bivariate abstract fractional calculus high order monotone (constrained) approximation theory by pseudo-polynomials of Caputo type, and then we apply our results to bivariate Prabhakar fractional Calculus, bivariate generalized non-singular fractional calculus, and bivariate parametrized Caputo-Fabrizio non-singular fractional calculus. Next, we build the related necessary fractional calculi background.

5.2 Bivariate Right Side Fractional Calculi Here we need to be very specific in preparation for our main results.

5.2.1 Bivariate Right Abstract Fractional Calculus Let h1 , h2 , v1 ,v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ / Z+ : C r,p [−1, 1]2 . Here h1 ≤ i ≤ v1 , h2 ≤ j ≤ v2 . Let α1i , α2j ≥ 0, α1i , α2j ∈ α1i = i, α2j = j, i = 0, 1, ..., r; j = 0, 1, ..., p, ( · is the ceiling of number), α10 = 0, α20 = 0. Consider also the integrable functions k1i := Kα1i , k2j := Kα2j : [0, 2] → R+ , i = 0, 1, ..., r; j = 0, 1, ..., p. We consider right side Caputo type bivariate fractional partial deriva  the abstract tive of orders α1i , α2j k1i (α1i ,α2j ) f k2j D1−

1 1 (x) := (−1)

k1i (t1 − x1 ) k2j (t2 − x2 )

i+j x1 x2

∂ i+j f (t1 , t2 ) j

∂t1i ∂t2

dt1 dt2 , (5.34)

∀ x = (x1 , x2 ) ∈ [−1, 1]2 . We set k10 (0,0) k20 D1− f k1i (i,j) k2j D1− f

(x) := (−1)i+j

(x) := f (x) ,

∂ i+j f (x1 ,x2 ) , j ∂x1i ∂x2

(5.35) ∀ x = (x1 , x2 ) ∈ [−1, 1]2 .

5.2 Bivariate Right Side Fractional Calculi

105

We also set k1i (i,α2j ) f k2j D1−

1 (x) := (−1)

k2j (t2 − x2 )

j

∂ i+j f (x1 , t2 )

k1i (α1i ,j) k2j D1− f

1 (x) := (−1)

k1i (t1 − x1 )

i

dt2 ,

(5.36)

dt1 ,

(5.37)

dt2 ,

(5.38)

∂ i f (t1 , x2 ) dt1 , ∂t1i

(5.39)

j

∂x1i ∂t2

x2

∂ i+j f (t1 , x2 )

x1

j

∂t1i ∂x2

and in particular we define: k10 (0,α2j ) f k2j D1−

1 (x) := (−1)

k2j (t2 − x2 )

j

∂ j f (x1 , t2 ) j

∂t2

x2

k1i (α1i ,0) f k20 D1−

1 (x) := (−1)i

k1i (t1 − x1 ) x1

∀ x = (x1 , x2 ) ∈ [−1, 1]2 . We will assume that 1 kihi (z) dz ≥ 1, when hi = 0,

(5.40)

0

where i = 1, 2.     In [3], we got that 0 < h1 − α1h1 + 1 , h2 − α2h2 + 1 ≤ 1, where is the gamma function, and there it is kihi (z) =

z hi −αihi −1  , i = 1, 2; ∀ z ∈ [0, 2] , 

hi − αihi

and (5.40) is fulfilled.

5.2.2 About Right Bivariate Fractional Calculus We consider the Prabhakar function (also known as the three parameter Mittag-Leffler function), (see [10], [13, p. 97])

106

5 Right Side Abstract Bivariate Monotone … γ

Eα,β (z) =

∞  k=0

(γ )k zk , k! (αk + β)

(5.41)

0 where α, β > 0, γ ∈ R, z ∈ R, and (γ )k = γ (γ + 1) ... (γ + k − 1) , it is Eα,β (z) = 1 .

(β) Let a, b, c, d ∈ R, a < b, c < d ; f ∈ C N1 ,N2 ([a, b] × [c, d ]), ρi , μi > 0; γi , ωi ∈ / N; i = 1, 2. R; Ni = μi , μi ∈ We define the bivariate Prabhakar-Caputo right partial fractional derivative of orders (μ1 , μ2 ) as follows (x = (x1 , x2 ) ∈ [a, b] × [c, d ]):

#

C

(−1)

$ (γ ,γ ) D(ρ11 ,ρ22 ),(μ1 ,μ2 ),(ω1 ,ω2 ),(b−,d −) f (x) =

N1 +N2

b d

(t1 − x1 )N1 −μ1 −1 (t2 − x2 )N2 −μ2 −1

(5.42)

x1 x2

% & −γ % & ∂ N1 +N2 f (t1 , t2 ) −γ Eρ1 ,N1 1 −μ1 ω1 (t1 − x1 )ρ1 Eρ2 ,N2 2 −μ2 ω2 (t2 − x2 )ρ2 dt1 dt2 , ∂t1N1 ∂t2N2 with #

C

#

C

$ (γ ,γ ) D(ρ11 ,ρ22 ),(0,0),(ω1 ,ω2 ),(b−,d −) f (x) := f (x) ;

(γ ,γ ) D(ρ11 ,ρ22 ),(N1 ,N2 ),(ω1 ,ω2 ),(b−,d −) f

$

(5.43) (x) :=

N1 +N2 (−1)N1 +N2 ∂ N1f (xN12,x2 ) , ∂x1 ∂x2

where N1 , N2 ∈ N, etc. For the related univariate theory see [5, 18].

5.2.3 About Right Generalized Non-singular Fractional Calculus Here we use the multivariate analogue  of  generalized Mittag-Leffler function, see [19], defined for λ, γj , ρj , zj ∈ C, Re ρj > 0 (j = 1, ..., m) in terms of a multiple series of the form: ( γj ) (γ ,...,γ ) E ρ ,λ (z1 , ..., zm ) = E(ρ11,...,ρmm ),λ (z1 , ..., zm ) = ( j)

5.2 Bivariate Right Side Fractional Calculi

107

∞ 

(γ1 )k1 ... (γm )km z1k1 ...zmkm   , m k1 !...km ! ' k1 ,...,km =0

λ+ kj ρj

(5.44)

j=1

  where γj kj is the Pochhammer symbol. By [22, p. 157], (5.44) converges for   Re ρj > 0, j = 1, ..., m. & (γ ,...,γ m ) % ωθ1 tθρθ , ..., ωθm tθρθ , denoted by In particular we will use θ = 1, 2; E(ρθθ 1,...,ρθθ),λ θ   (γθ j ) % ρ ρ & E(ρθ ),λθ ωθ1 tθ θ , ..., ωθm tθ θ , where 0 < ρθ < 1, tθ ≥ 0, λθ > 0, γθj ∈ R with γθj kθ j     := γθj γθj + 1 ... γθj + kθj − 1 , ωθj ∈ R − {0}, for j = 1, ..., m.   / N and μθ = Nθ ∈ N; θ = 1, 2. Let f ∈ C N1 ,N2 [−1, 1]2 , 0 < μθ ∈ We define the bivariate Caputo type generalized right partial fractional derivative with non-singular kernel of order (μ1 , μ2 ), as follows: (μ1 D1−

,μ2 )

A (1 − n1 + μ1 , 1 − N2 + μ2 ) (γ1j )(ω1j )CA (μ1 ,μ2 ),(λ1 ,λ2 ) D1− f (x) := γ ω f (x) := ( 2j )( 2j ) (N1 − μ1 ) (N2 − μ2 )

(−1)N1 +N2

 −ωθ1 (1 − Nθ + μθ ) ( γθ j ) E(1−Nθ +μθ ),λθ (tθ − xθ )1−Nθ +μθ N − μ θ θ θ=1

1 1 ( 2 x1 x2

 N1 +N2 f (t1 , t2 ) −ωθm (1 − Nθ + μθ ) 1−Nθ +μθ ∂ , ..., dt1 dt2 , (tθ − xθ ) N1 N2 Nθ − μθ ∂t1 ∂t2

(5.45)

∀ x = (x1 , x2 ) ∈ [−1, 1]2 , where A := A (1 − n1 + μ1 , 1 − N2 + μ2 ) is a normalizing constant. Without loss of generality we assume that A > 0. N1 +N2 (0,0) (N1 ,N2 ) We set D1− f = f , D1− f = (−1)N1 +N2 ∂ N1 Nf2 , when N1 , N2 ∈ N, etc. ∂x1 ∂x2

For the univariate theory see the related [4, 6, 9].

5.2.4 Bivariate Right Parametrized Caputo-Fabrizio Type Non-singular Kernel Left Partial Fractional Derivative of Orders (µ1 , µ2 )   Let f ∈ C N1 ,N2 [−1, 1]2 , N  μ1 , μ2 > 0, μθ = Nθ ∈ N; ωθ < 0; θ = 1, 2. It is given by (μ1 ,μ2 ) CF f (ω1 ,ω2 ) D1−

(x) :=

(−1)N1 +N2 (N1 − μ1 ) (N2 − μ2 )

108

5 Right Side Abstract Bivariate Monotone …

 N1 +N2  1 1 ( 2  ∂ f (t1 , t2 ) (1 − Nθ + μθ ) ωθ exp − dt1 dt2 , (5.46) (tθ − xθ ) N 1 Nθ − μθ ∂t1 ∂t2N2 θ=1

x1 x2

∀ x = (x1 , x2 ) ∈ [−1, 1]2 , and (0,0) (N1 ,N2 ) CF CF f = (−1)N1 +N2 (ω1 ,ω2 ) D1− f = f , (ω1 ,ω2 ) D1− For the univariate case see [6, 15]. We make

∂ N1 +N2 f N N ∂x1 1 ∂x2 2

, etc.

Remark 5.9 Right Fractional Calculi Sects. 5.2.2–5.2.4 are special cases of the right abstract fractional calculus Sect. 5.2.2. The abstract important condition (5.40) is fulfilled by: in Sect. 5.2.2 for large enough ωθ > 0, θ = 1, 2; in Sect. 5.2.3 for small enough ωθj < 0, θ = 1, 2; j = 1, ..., m; and in Sect. 5.2.4 for small enough ωθ < 0, θ = 1, 2. For details see [6] and chap. 1.

5.3 Main Result We present the following right bivariate abstract fractional monotone (constrained) approximation result. Theorem 5.10 Leth1 , h2 , v1, v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ 2 (r,p) having a mixed modulus of smoothness p and let f ∈ C r,p  (r,p)  [−1, 1] , with f ; δ1 , δ2 , δ1 , δ2 > 0 there, s, q ∈ N. Let αij (x, y), i = h1 , h1 + 1, ..., v1 ; ωs,q f j = h2 , h2 + 1, ..., v2 be real-valued functions, defined and bounded in [−1, 1]2 and suppose αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout [−1, 0]2 . Assume that h1 , h2 are even. Here n, m ∈ N : n ≥ max {4 (r + 1) , r + s}, m ≥ max {4 (p + 1) , p + q}. Set  −1  α  (5.47) sup lij := h1 h2 (x, y) αij (x, y) < ∞ (x,y)∈[−1,1]2

  for all h1 ≤ i ≤ v1 , h2 ≤ j ≤ v2 . Let 0 < α1i , α2j ∈ / N with α1i = i, α2j = j, j = 1, ..., r; j = 1, ..., p ( · is the ceiling of the number) and α10 = α20 = 0. Consider the abstract right fractional bivariate differential operator L∗ :=

v2 v1  

αij (x, y)

k1i (α1i ,α2j ) . k2j D1−

(5.48)

i=h1 j=h2

Assume L∗ f (x, y) ≥ 0, on [−1, 0]2 .There exists a pseudo-polynomial of degree ≤ (n, m) ∗ ∗ := Qn,m Qn,m (f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) ∗ such that L∗ Qn,m (x, y) ≥ 0, on [−1, 0]2 .

5.3 Main Result

109

We set

2 λ1i :=

2 k1i (z) dz, λ2j :=

0

k2j (z) dz,

(5.49)

0

for i = 1, ..., r; j = 1, ..., p. Set also λ10 = λ20 = 1. Furthermore it holds: (1) if (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 , then  k1i (α1i ,α2j ) f − k2j D1−



k1i (α1i ,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

≤

   C (r,p) 1 1 , λ1i λ2j r−i p−j ωs,q f ; , n m n m

(5.50)

(2) if (1, 1) ≤ (i, j) ≤ (h1 , h2 ), then    Cωs,q f (r,p) ; 1n , m1 ≤ nr−v1 mp−v2 ⎧⎛ v1 v2 ⎫ ⎞ ' ' ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎜ i =h j =h li∗ j∗ λ1i∗ λ2j∗ ⎟  ⎬  ⎜∗ 1 ∗ 2 ⎟ k1i (α1i ,α2j )  h1 h2  , x y  + λ λ ⎜ ⎟ k2j D1− 1i 2j ⎪ ⎪ ∞,[−1,1]2 ⎝ ⎠ h1 !h2 ! ⎪ ⎪ ⎪ ⎪ ⎩ ⎭  k1i (α1i ,α2j ) f − k2j D1−



k1i (α1i ,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

(5.51)

(3) it holds   f − Q∗  n,m ∞,[−1,1]2

⎤ ⎡ v1 v2 ' ' li∗ j∗ λ1i∗ λ2j∗  (r,p) 1 1  ⎢ ⎥  Cωs,q f ; n , m ⎢ i∗ =h1 j∗ =h2 ⎥ + 1 ≤ ⎥ , (5.52) ⎢ r−v p−v 1 2 ⎦ ⎣ n m h1 !h2 !

(4) when i = 0, j = 1, ..., h2 , we get  k10 (0,α2j ) k2j D1− f − ⎡⎛



k10 (0,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

   Cωs,q f (r,p) ; 1n , m1 ≤ nr−v1 mp−v2 ⎤

⎞ l λ λ  ⎢⎜ i =h j =h i∗ j∗ 1i∗ 2j∗ ⎟  ⎥ ⎢⎜ ∗ 1 ∗ 2 ⎥ ⎟ k10 (0,α2j ) h2  + λ2j ⎥ , ⎢⎜ ⎟ k2j D1− y  ∞,[−1,1] ⎣⎝ ⎦ ⎠ h1 !h2 ! v1 v2 ' '

(5.53)

110

5 Right Side Abstract Bivariate Monotone …

and (5) the case of j = 0, i = 1, ..., h1 follows, it holds:  k1i (α1i ,0) k20 D1− f − ⎡⎛



k1i (α1i ,0) ∗  Qn,m  k20 D1− ∞,[−1,1]2

⎞

v1 v2 ' '

   Cωs,q f (r,p) ; 1n , m1 ≤ nr−v1 mp−v2 ⎤

⎢⎜ i =h j =h li∗ j∗ λ1i∗ λ2j∗ ⎟  ⎥  ⎢⎜ ∗ 1 ∗ 2 ⎥ ⎟ k1i (α1i ,0) h1  + λ1i ⎥ . ⎢⎜ ⎟ k20 D1− x  ∞,[−1,1] ⎣⎝ ⎦ ⎠ h1 !h2 !

(5.54)

Proof By Corollary 5.3 there exists Qn,m ≡ Qn,m (f ) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ) such that

 (i,j)  f − Q(i,j)  n,m

∞

   C (r,p) 1 1 , ≤ r−i p−j · ωs,q f ; , n m n m

(5.55)

  for all (0, 0) ≤ (i, j) ≤ (r, p), while Qn,m ∈ C r,p [−1, 1]2 . Here  C depends only on {4 r, p, s, q, where n ≥ max {4 (r + 1) , r + s} and m ≥ max + 1) , p + q}, with (p   r, p ∈ Z+ , s, q ∈ N, f ∈ C r,p [−1, 1]2 . (r,p) Indeed by [12] we have that Qn,m is continuous on [−1, 1]2 . We observe the following (i = 1, ..., r; j = 1, ..., p)  k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D1−

k1i (α1i ,α2j ) Qn,m k2j D1−

  (x1 , x2 ) =

  1 1  ∂ i+j f (t1 , t2 ) i+j (−1) k1i (t1 − x1 ) k2j (t2 − x2 ) dt1 dt2 −  j ∂t1i ∂t2  x1 x2

1 1 k1i (t1 − x1 ) k2j (t2 − x2 )

(−1)i+j x1 x2

∂ i+j Qn,m (t1 , t2 ) j

∂t1i ∂t2

   dt1 dt2  = 

(5.56)

 1 1       i+j i+j   ∂ f , t Q , t ∂ (t ) (t ) 1 2 n,m 1 2  k1i (t1 − x1 ) k2j (t2 − x2 ) − dt1 dt2  ≤  i j i j ∂t1 ∂t2 ∂t1 ∂t2   x1 x2

1 1 x1 x2

   ∂ i+j f (t , t ) ∂ i+j Q (t , t )  1 2 n,m 1 2   k1i (t1 − x1 ) k2j (t2 − x2 )  −  dt1 dt2 ≤ j  ∂t i ∂t j  ∂t i ∂t 1

2

1

2

5.3 Main Result

111

⎛ 1 1 ⎞   ⎝ k1i (t1 − x1 ) k2j (t2 − x2 ) dt1 dt2 ⎠

   C (r,p) 1 1 f = ω ; , s,q nr−i mp−j n m

x1 x2

⎛ ⎝

1

⎞⎛ 1 ⎞  k1i (t1 − x1 ) dt1 ⎠ ⎝ k2j (t2 − x2 ) dt2 ⎠

x1

   C (r,p) 1 1 ωs,q f ; , = nr−i mp−j n m

x2

⎞ ⎛ 1−x ⎞ ⎛ 1−x  1  2 ⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

   C (r,p) 1 1 f , ≤ ω ; s,q nr−i mp−j n m

(5.57)

0

⎛ 2 ⎞⎛ 2 ⎞   ⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

   C (r,p) 1 1 . ωs,q f ; , nr−i mp−j n m

0

We have proved that  k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D1−

k1i (α1i ,α2j ) Qn,m k2j D1−

⎛ 2 ⎞⎛ 2 ⎞   ⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

  (x1 , x2 ) ≤

(5.58)

   C (r,p) 1 1 , ωs,q f ; , nr−i mp−j n m

0

∀ (x1 , x2 ) ∈ [−1, 1]2 ; i = 1, ..., r; j = 1, ..., p. So we have proved that there exists Qn,m such that  k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

⎛ 2 ⎞⎛ 2 ⎞   ⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

   Qn,m 

∞,[−1,1]2

≤

(5.59)

   C (r,p) 1 1 , , f ω ; s,q nr−i mp−j n m

0

i = 1, ..., r; j = 1, ..., p. We call

2

λ1i :=

2 k1i (z) dz, λ2j :=

0

k2j (z) dz, 0

for i = 1, ..., r; j = 1, ..., p, as in (5.49). We also set λ10 = λ20 = 1. Thus the following inequality is valid in general:

(5.60)

112

5 Right Side Abstract Bivariate Monotone …

 k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

   Qn,m 

∞,[−1,1]2

≤

(5.61)

   C (r,p) 1 1 , λ1i λ2j r−i p−j ωs,q f ; , n m n m for i = 0, 1, ..., r; j = 0, 1, ..., p. Define ρn,m

⎤ ⎡    v1  v2   1 1 ⎣ lij λ1i λ2j ni−r mj−p ⎦ . :=  Cωs,q f (r,p) ; , n m

(5.62)

i=h1 j=h2

∗ (I) Suppose, throughout [−1, 0]2 , αh1 h2 (x, y) ≥ α > 0. Let Qn,m (x, y) ∈ (Pn ⊗ C 2 ([−1, 1]) + C ([−1, 1]) ⊗ Pm ), (x, y) ∈ [−1, 1] , as in (5.61), so that

   k1i (α1i ,α2j ) xh1 yh2  D f (x, y) + ρn,m − k2j 1− h !h ! 1

2

k1i (α1i ,α2j ) ∗ Qn,m k2j D1−

  (x, y) 

∞,[−1,1]2

   C (r,p) 1 1 λ1i λ2j r−i p−j ωs,q f =: Tij , ; , n m n m

≤

(5.63)

for i = 0, 1, ..., r; j = 0, 1, ..., p. If (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 we get from the last 

 k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

≤

   C (r,p) 1 1 , λ1i λ2j r−i p−j ωs,q f ; , n m n m

(5.64)

proving (5.50). If (0, 0) ≤ (i, j) ≤ (h1 , h2 ), we get that  k1i (α1i ,α2j ) ρn,m  D f + k2j 1− h1 !h2 !

k1i (α1i ,α2j ) k2j D1−

 h1 h2  x y −

k1i (α1i ,α2j ) ∗ Qn,m k2j D1−

≤ Tij .

∞,[−1,1]2

(5.65)

That is for (1, 1) ≤ (i, j) ≤ (h1 , h2 ), we have  k1i (α1i ,α2j ) f − k2j D1−

  (x, y) 



k1i (α1i ,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

≤

5.3 Main Result

113

 ρn,m  k1i (α1i ,α2j )  h1 h2  x y  + Tij = k2j D1− ∞,[−1,1]2 h1 !h2 ! ⎤   ⎡ v1 v2     Cωs,q f (r,p) ; 1n , m1 ⎣ li∗ j∗ λ1i∗ λ2j∗ ni∗ −r mj∗ −p ⎦ h1 !h2 ! i∗ =h1 j∗ =h2

  k1i (α1i ,α2j )  h1 h2  x y  k2j D1−

∞,[−1,1]2

+ λ1i λ2j

   C (r,p) 1 1 f , = ω ; s,q nr−i mp−j n m

(5.66)

⎧ ⎤ ⎡  ⎨ v1  v2  1 1 1  ⎣ Cωs,q f (r,p) ; , li∗ j∗ λ1i∗ λ2j∗ ni∗ −r mj∗ −p ⎦ n m ⎩ h1 !h2 ! i∗ =h1 j∗ =h2

  k1i (α1i ,α2j )  h1 h2  x y  k2j D1−

∞,[−1,1]2

λ1i λ2j + r−i p−j n m

. ≤

⎧⎛ v1 v2 ⎞ ' ' ⎪ l λ λ  (r,p) 1 1  ⎪ ⎪ i j 1i 2j ∗ ∗ ∗ ∗ ⎟  Cωs,q f ; n , m ⎨⎜ ⎜ i∗ =h1 j∗ =h2 ⎟ ⎜ ⎟ r−v p−v 1 2 ⎪ ⎝ ⎠ n m h1 !h2 ! ⎪ ⎪ ⎩   k1i (α1i ,α2j )  h1 h2  x y  k2j D1−

(5.67)

. ∞,[−1,1]2

+ λ1i λ2j ,

proving (5.51). If i = j = 0, from (5.63) we obtain        C (r,p) 1 1 f + ρn,m xh1 yh2 − Q∗  , ≤ r p ωs,q f ; , n,m   h1 !h2 ! nm n m ∞,[−1,1]2 and

     C ρn,m (r,p) 1 1 f − Q∗  f + , = ≤ ω ; s,q n,m ∞,[−1,1]2 h1 !h2 ! nr mp n m ⎤   ⎡ v1 v2    Cωs,q f (r,p) ; 1n , m1 ⎣ li∗ j∗ λ1i∗ λ2j∗ ni∗ −r mj∗ −p ⎦ h1 !h2 ! i∗ =h1 j∗ =h2

+

   C (r,p) 1 1 f , = ω ; s,q nr mp n m

(5.68)

114

5 Right Side Abstract Bivariate Monotone …

⎤ ⎡   v1  v2  1 1 1 1  ⎣ Cωs,q f (r,p) ; , li∗ j∗ λ1i∗ λ2j∗ ni∗ −r mj∗ −p + r p ⎦ ≤ (5.69) n m h1 !h2 ! nm i∗ =h1 j∗ =h2

⎡ v1 v2 ⎤ ' ' li∗ j∗ λ1i∗ λ2j∗  (r,p) 1 1  ⎢ ⎥  Cωs,q f ; n , m ⎢ i∗ =h1 j∗ =h2 ⎥ + 1 ⎢ ⎥, ⎣ ⎦ nr−v1 mp−v2 h1 !h2 !

proving (5.52). Next case is for i = 0, j = 1, ..., h2 , from (5.63) we get  k10 (0,α2j ) ρn,m xh1  D f + k2j 1− h1 !h2 !

k10 (0,α2j ) h2 y k2j D1−



−

 k10 (0,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

≤

(5.70)

   C (r,p) 1 1 λ2j r p−j ωs,q f . ; , nm n m Therefore we have that  k10 (0,α2j ) k2j D1− f −



k10 (0,α2j ) ∗  Qn,m  k2j D1− ∞,[−1,1]2

≤

    C ρn,m  k10 (0,α2j ) h2  (r,p) 1 1 = y  + λ2j r p−j ωs,q f ; ,  D ∞,[−1,1] h1 !h2 ! k2j 1− nm n m ⎤   ⎡ v1 v2     Cωs,q f (r,p) ; 1n , m1 ⎣ li∗ j∗ λ1i∗ λ2j∗ ni∗ −r mj∗ −p ⎦ h1 !h2 ! i∗ =h1 j∗ =h2

  k10 (0,α2j ) h2  k2j D1− y 

∞,[−1,1]

+ λ2j

⎡/  ⎢ 1 1 ⎢ ⎢  Cωs,q f (r,p) ; , n m ⎢ ⎣

   C (r,p) 1 1 f , = ω ; s,q nr mp−j n m

" v1 v2   ' ' i∗ −r j∗ −p li∗ j∗ λ1i∗ λ2j∗ n m

i∗ =h1 j∗ =h2

  k10 (0,α2j ) h2  k2j D1− y 

h1 !h2 !

∞,[−1,1]

+

λ2j r n mp−j

 ≤

(5.71)

5.3 Main Result

115

⎡  ⎢  Cωs,q f (r,p) ; 1n , m1 ⎢ ⎢ ⎢ nr−v1 mp−v2 ⎣

v1 v2 ' '

i∗ =h1 j∗ =h2

 li∗ j∗ λ1i∗ λ2j∗

h1 !h2 ! 

  k10 (0,α2j ) h2  k2j D1− y 

+ λ2j ,

∞,[−1,1]

proving (5.53). The case of j = 0, i = 1, ..., h1 , is met similarly as in (5.71). Namely we get  k1i (α1i ,0) k20 D1− f −



k1i (α1i ,0) ∗  Qn,m  k20 D1− ∞,[−1,1]2

⎡⎛

≤ ⎞

v1 v2 ' '

li∗ j∗ λ1i∗ λ2j∗ ⎟   ⎜  Cωs,q f (r,p) ; 1n , m1 ⎢ ⎢⎜ i∗ =h1 j∗ =h2 ⎟ ⎢⎜ ⎟ ⎣⎝ ⎠ nr−v1 mp−v2 h1 !h2 !   k1i (α1i ,0) h1  k20 D1− x 

∞,[−1,1]

(5.72)

 + λ1i ,

proving (5.54). So if (x, y) ∈ [−1, 0]2 , we can write  ∗  αh−1 (x, y) L∗ Qn,m (x, y) = αh−1 (x, y) L∗ (f (x, y)) + 1 h2 1 h2 ρn,m h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

i=h1 j=h2



k1i (α1i ,α2j ) ∗ Qn,m k2j D1−

(by L∗ f ≥ 0)

v1  v2  h1 h2   x y + αh−1 (x, y) αij (x, y) · 1 h2

(x, y) −

(5.63)

≥

k1i (α1i ,α2j ) f k2j D1−

ρn,m h1 !h2 !

ρn,m (x, y) − h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

k1i (α1i ,α2j ) k2j D1−

 h1 h2  x y −

⎛ ⎞   v1  v2  i−r j−p ⎠  (r,p) 1 1 ⎝ = lij λ1i λ2j n m ; , Cωs,q f n m i=h1 j=h2

  h1 h2  x y (5.73)

116

5 Right Side Abstract Bivariate Monotone …

⎛k

1h1

ρn,m ⎝ ⎡# ⎢ ρn,m ⎣

k2h2

 (α1h ,α2h )  D1− 1 2 xh1 yh2 h1 !h2 !

k1h1 (α1h1 ,0) h x1 k20 D1−

$#

⎞ − 1⎠ =

(0,α2h2 ) h2 k10 y k2h2 D1−

(5.74)

⎤

$

⎥ − 1⎦ =: ϕ.

h1 !h2 !

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ϕ = 0. If h1 = 0, and h2 = 0, then ⎡ ϕ = ρn,m ⎣ ⎡ ρn,m ⎣(−1)h2

1

(0,α2h2 ) h2 k10 y k2h2 D1− h2 !

⎤ − 1⎦ =

⎛ 1−y ⎞  (h is even) k2h2 (t − y) dt − 1⎦ 2 = ρn,m ⎝ k2h2 (z) dz − 1⎠ ≥ (5.75) ⎤

y

0

⎛ 1 ⎞  ρn,m ⎝ k2h2 (z) dz − 1⎠ ≥ 0, 0

by the assumption (5.40). Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡⎛ 1 ⎞⎛ 1 ⎞ ⎤   (h ,h are even) ϕ 1 2= ρn,m ⎣⎝ k1h1 (t − x) dt ⎠ ⎝ k2h2 (t − y) dt ⎠ − 1⎦ = x

(5.76)

y

⎡⎛ 1−x ⎞ ⎛ 1−y ⎞ ⎤   ρn,m ⎣⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠ − 1⎦ ≥ 0

0

⎡⎛ 1 ⎞⎛ 1 ⎞ ⎤   ρn,m ⎣⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠ − 1⎦ ≥ 0, 0

0

by the assumption (5.40). So in all four cases we get that  ∗  L∗ Qn,m (x, y) ≥ 0, ∀ (x, y) ∈ [−1, 0]2 .

(5.77)

5.3 Main Result

117

∗ (II) Suppose, throughout [−1, 0]2 , αh1 h2 (x, y) ≤ β < 0. Let Qn,m (x, y) ∈ (Pn ⊗ C ([−1, 1]) + C ([−1, 1]) ⊗ Pm ), (x, y) ∈ [−1, 1]2 , as in (5.61), so that

   k1i (α1i ,α2j ) xh1 yh2  D f y) − ρ − (x, n,m k2j 1− h !h ! 1

k1i (α1i ,α2j ) ∗∗ Qn,m k2j D1−

2

  (x, y) 

∞,[−1,1]2

   C (r,p) 1 1 f , , ω ; s,q nr−i mp−j n m

λ1i λ2j

≤

(5.78)

for i = 0, 1, ..., r, j = 0, 1, ..., p. Similarly, we get as in the first case ≥ α > 0, the inequalities of simultaneous fractional convergence, see {(5.51), (5.67)}, {(5.52), (5.69)}, {(5.53), (5.71)}, {(5.54), (5.72)}. So if (x, y) ∈ [−1, 0]2 we can write  ∗  αh−1 (x, y) L∗ Qn,m (x, y) = αh−1 (x, y) L∗ (f (x, y)) − 1 h2 1 h2 ρn,m h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

v1  v2  h1 h2   x y + αh−1 (x, y) αij (x, y) · 1 h2

(5.79)

i=h1 j=h2



k1i (α1i ,α2j ) ∗ Qn,m k2j D1−

(x, y) −

(by L∗ f ≥ 0) (5.78)

≤ −

⎛ ⎝

v1  v2  i=h1 j=h2

k1i (α1i ,α2j ) f k2j D1−

ρn,m h1 !h2 !

k1i (α1i ,α2j ) k2j D1−

  h1 h2  x y

 h1 h2  x y +

  1 1 = lij λ1i λ2j nr−i mp−j ⎠  Cωs,q f (r,p) ; , n m

ρn,m ⎣1 −

⎢ ρn,m ⎣1 −

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

ρn,m h1 !h2 !

⎞

⎡

⎡

(x, y) +

#k

1h1

k20

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

h1 !h2 !

(α1h ,0) D1− 1 xh1

 h h ⎤ x 1y 2 ⎦=

$#

(0,α2h2 ) h2 k10 y k2h2 D1−

h1 !h2 !

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ψ = 0.

$⎤ ⎥ ⎦ =: ψ.

(5.80)

118

5 Right Side Abstract Bivariate Monotone …

If h1 = 0, and h2 = 0, then ⎡ ψ = ρn,m ⎣1 −

(0,α2h2 ) h2 k10 y k2h2 D1− h2 !

⎤

⎡

⎦ = ρn,m ⎣1 −

1

⎤ k2h2 (t − y) dt ⎦ =

(5.81)

y

⎡

⎡ ⎤ ⎤ 1−y 1 ρn,m ⎣1 − k2h2 (z) dz ⎦ ≤ ρn,m ⎣1 − k2h2 (z) dz ⎦ ≤ 0, 0

0

see (5.40). Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡

⎛

ψ = ρn,m ⎣1 − ⎝

1

⎞⎛ k1h1 (t − x) dt ⎠ ⎝

x

1

⎞⎤ k2h2 (t − y) dt ⎠⎦ =

y

⎡

⎛ 1−x ⎞ ⎛ 1−y ⎞⎤   ρn,m ⎣1 − ⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠⎦ ≤ 0

⎡

⎛

ρn,m ⎣1 − ⎝

1

(5.82)

0

⎞⎛ 1 ⎞⎤  k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠⎦ ≤ 0,

0

0

by the assumption (5.40). So in all four cases we proved again that  ∗  L∗ Qn,m (x, y) ≥ 0, ∀ (x, y) ∈ [−1, 0]2 . The proof is complete.

(5.83) 

Conclusion 5.11 Clearly Theorem 5.10 generalizes Theorem 5.8 to many right side fractional calculi, opening new avenues of fractional research activity. The approxi∗ depends on f , ρn,m , h1 , h2 ; which ρn,m depends on  C mating pseudo-polynomial Qn,m (which depends on r, p, s, q), f , n, m, lij , λ1i , λ2j ; and which: λ1i depends on k1i and ∗ depends on the type of bivariate right side fractional λ2j depends on k2j . That is Qn,m calculus we use. Consequently, Theorem 5.10 is valid at least for the following important bivariate right fractional linear differential operators:

5.3 Main Result

(1) L∗1 :=

119

v1  v2 

αij (x, y)

0

i=h1 j=h2

C

1 (γ ,γ ) D(ρ1 ,ρ2 ), α ,α ,(ω ,ω ),(1−,1−) , 1 2 ( 1i 2j ) 1 2

(5.84)

where ρ1 , ρ2 > 0, γ1 , γ2 < 0, and ω1 , ω2 > 0 large enough (from right bivariate Prabhakar fractional calculus, see (5.42)); (2) L∗2

:=

v1  v2 

0 α ,α 1 ( 1i 2j ) , αij (x, y) D1−

(5.85)

i=h1 j=h2

(see (5.45)) where θ = 1, 2; γθj > 0, j = 1, ..., m; λθ = 1, 0 < ρθ < 1; and small enough ωθj < 0, j = 1, ..., m (from right bivariate generalized non-singular fractional calculus); and (3) L∗3 :=

v1  v2 

αij (x, y)

0

(α1i ,α2j ) CF (ω1 ,ω2 ) D1−

1

,

(5.86)

i=h1 j=h2

(see (5.46)) for small enough ω1 , ω2 < 0 (from right bivariate parametrized Caputo-Fabrizio non-singular kernel fractional calculus). Our developed right bivariate abstract fractional monotone approximation theory by pseudopolynomials with its applications, involves weaker conditions than the one with ordinary partial derivatives, see Theorem 5.4, and can manage many diverse general cases in a multitude of complex settings and environments.

References 1. Anastassiou, G.A.: Monotone approximation by pseudopolynomials. In: Approximation Theory, pp. 5–11. Academic Press, New York (1991) 2. Anastassiou, G.A.: Bivariate monotone approximation. Proc. Amer. Math. Soc. 112(4), 959– 964 (1991) 3. Anastassiou, G.A.: Bivariate right fractional pseudo-polynomial monotone approximation. In: Intelligent Mathematics II: Applied Mathematics and Approximation Theory, pp. 15–31. AMAT, Ankara, May 2015, Springer, New York (2016) 4. Anastassiou, G.A.: Multiparameter fractional differentiation with non singular kernel. Issues. Anal. (2021) (accepted) 5. Anastassiou, G.A.: Foundations of generalized Prabhakar-Hilfer fractional calculus with applications. Cubo (2021) (accepted) 6. Anastassiou, G.A.: Univariate simultaneous high order abstract fractional monotone approximation with applications (2021) (submitted) 7. Anastassiou, G.A.: Abstract bivariate right fractional pseudo-polynomial monotone constrained approximation and applications (2021) (submitted)

120

5 Right Side Abstract Bivariate Monotone …

8. Anastassiou, G.A., Shisha, O.: Monotone approximation with linear differential operators. J. Approx. Theory 44, 391–393 (1985) 9. Atangana, A., Baleanu, D.: New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016) 10. Giusti, A., et al.: A practical Guide to Prabhakar fractional calculus. Fract. Calculus App. Anal. 23(1), 9–54 (2020) 11. Gonska, H.H.: Simultaneously approximation by algebraic blending functions. In: Alfred Haar memorial conference, Budapest, vol 49, pp. 363–382. Coloquia Mathematica Soc. Janos Bolyai. North-Holand, Amsterdam (1985) 12. Gonska, H.H.: Personal communication with author (2014) 13. Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Heidelberg, New York (2014) 14. Iqbal, S., Krulic, K., Pecaric, J.: On an inequality of H.G. Hardy. J. Inequal. Appl. 2010(Article ID 264347), 23 p 15. Losada,J., Nieto, J.J.: Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 87–92 (2015) 16. Marchaud, A.: Differences et deerivees d’une fonction de deux variables. C.R. Acad. Sci. 178 1467–1470 (1924) 17. Marchaud, A.: Sur les derivees et sur les differences des fonctions de variables reelles. J. Math. Pures Appl. 6, 337–425 (1927) 18. Polito, F., Tomovski, Z.: Some properties of Prabhakar-type fractional calculus operators. Fract. Differ. Calculus 6(1), 73–94 (2016) 19. Saxena, R.K., Kalla, S.L.: Ravi Saxena. Multivariate analogue of generalized Mittag-Leffler function. Integr. Transf. Spec. Funct. 22(7), 533–548 (2011) 20. Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York (1981) 21. Shisha, O.: Monotone approximation. Pac. J. Math. 15, 667–671 (1965) 22. Srivastava, H.M., Daoust, M.C.: A note on the convergence of Kompe’ de Feriet’s double hypergeometric series. Math. Nachr. 53, 151–159 (1972)

Chapter 6

Bivariate Polynomial Abstract Left and Right Fractional Monotone Constrained Approximation

  Let f ∈ C r,p [0, 1]2 , r, p ∈ N, and let L∗ be an abstract linear left or right fractional mixed partial differential operator such that L∗ (f ) ≥ 0, for all (x, y) in a critical region there exists of [0, 1]2 that depends on L∗ . Then   a sequentce of two-dimensional polynomials Qm1 ,m2 (x, y) with L∗ Qm1 ,m2 (x, y) ≥ 0 there, where m1 , m2 ∈ N such that m1 > r, m2 > p, so that f is approximated left or right abstract fractionally simultaneously and uniformly by Qm1 ,m2 on [0, 1]2 . This restricted left or right abstract fractional approximation is achieved quantitatively by the use of a suitable integer partial derivatives two-dimensional first modulus of continuity. This monotone constrained fractional approximation applies to a wide range of Caputo type fractional calculi of singular or non-singular kernels. It follows [4].

6.1 Introduction The topic of monotone approximation started in [8] has become a major trend in approximation theory. A typical problem in this subject is: given a positive integer k, approximate a given function whose kth derivative is ≥ 0 by polynomials having this property. In [5] the authors replaced the kth derivative with a linear differential operator of order k. We mention this motivating result. Theorem 6.1 Let h, k, p be integers, 0 ≤ h ≤ k ≤ p and let f be areal function, f (p) continuous in [−1, 1] with first modulus of continuity ω1 f (p) , x there. Let aj (x), j = h, h + 1, ..., k be real functions, defined and bounded on [−1, 1] and assume ah (x) is either ≥ some number α > 0 or ≤ some number β < 0 throughout [−1, 1]. Consider the operator

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2_6

121

122

6 Bivariate Polynomial Abstract Left and Right …

L=

k 



dj dxj

aj (x)

j=h



and suppose, throughout [−1, 1], L (f ) ≥ 0.

(6.1)

Then, for every integer n ≥ 1, there is a real polynomial Qn (x) of degree ≤ n such that L (Qn ) ≥ 0 throughout [−1, 1] and max |f (x) − Qn (x)| ≤ Cn

k−p

−1≤x≤1

  (p) 1 , ω1 f , n

where C is independent of n or f . We need

  Definition 6.2 (Stancu [9]) Let f ∈ C [0, 1]2 , [0, 1]2 = [0, 1] × [0, 1], where (x1 , y1 ) , (x2 , y2 ) ∈ [0, 1]2 and δ1 , δ2 ≥ 0. The first modulus of continuity of f is defined as follows: ω1 (f , δ1 , δ2 ) =

sup

|x1 −x2 |≤δ1 |y1 −y2 |≤δ2

|f (x1 , y1 ) − f (x2 , y2 )| .

Definition 6.3 Let f be a real-valued function defined on [0, 1]2 and let m, n be two positive integers. Let Bm,n be the Bernstein (polynomial) operator of order (m, n) given by Bm,n (f ; x, y) =

      n m   i j m n , · f · · xi (1 − x)m−i yj (1 − y)n−j . i j m n i=0 j=0

(6.2) For integers r, s ≥ 0, we denote by f (r,s) the differential operator of order (r, s) given by ∂ r+s f (x, y) . f (r,s) (x, y) = ∂xr ∂ys We use Theorem 6.4 (Badea and Badea [6]) It holds that (k,l) (k,l)  − Bm,n f f ≤ t (k, l) · ∞

6.1 Introduction

123

 

 1 k (k − 1) l (l − 1) 1 ,√ + max , · f (k,l) ∞ , (6.3) ω1 f (k,l) ; √ m n m−k n−l where m > k ≥ 0, n > l ≥ 0 are integers, f is a real-valued function on [0, 1]2 such that f (k,l) is continuous, and t is a positive real-valued function on Z2+ = {0, 1, 2, ...}2 . Here ·∞ is the supremum norm on [0, 1]2 .   Denote C r,p [0, 1]2 := {f : [0, 1]2 → R; f (k,l) is continuous for 0 ≤ k ≤ r, 0 ≤ l ≤ p}. In [1] the author proved the following motivational result. Theorem 6.5 Let  h1 , h2, v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ C r,p [0, 1]2 . Let αi,j (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real-valued functions, defined and bounded in [0, 1]2 and suppose αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout [0, 1]2 . Consider the operator L=

v1  v2 

αij (x, y)

i=h1 j=h2

∂ i+j ∂xi ∂yj

(6.4)

and assume that throughout [0, 1]2 , L (f ) ≥ 0. Then for integers m, n with  m > r, n > p, there exists a polynomial Qm,n (x, y) of degree (m, n) such that L Qm,n (x, y) ≥ 0 throughout [0, 1]2 and (k,l) f − Q(k,l) m,n

∞

≤

Pm,n (L, f ) k,l + Mm,n (f ) , (h1 − k)! (h2 − l)!

(6.5)

all (0, 0) ≤ (k, l) ≤ (h1 , h2 ). Furthermore we get (k,l) f − Q(k,l) m,n

∞

k,l ≤ Mm,n (f ) ,

(6.6)

for all (h1 + 1, h2 + 1) ≤ (k, l) ≤ (r, p). Also (6.6) is true whenever 0 ≤ k ≤ h1 , h2 + 1 ≤ l ≤ p or h1 + 1 ≤ k ≤ r, 0 ≤ l ≤ h2 . Here k,l k,l ≡ Mm,n Mm,n (f ) ≡ t (k, l) ·

  

k (k − 1) l (l − 1) 1 1 , · f (k,l) ∞ , (6.7) + max ,√ ω1 f (k,l) ; √ m n m−k n−l and Pm,n ≡ Pm,n (L, f ) ≡

v1  v2  i=h1 j=h2

i,j lij · Mm,n ,

(6.8)

124

6 Bivariate Polynomial Abstract Left and Right …

where t is a positive real-valued function on Z2+ and lij ≡

sup (x,y)∈[0,1]2

−1 α h1 h2 (x, y) · αij (x, y) < ∞.

(6.9)

In [2] we extended Theorem 6.5 to the fractional level. Indeed there L is replaced by L∗ , a linear left Caputo fractional mixed partial differential operator. Now the monotonicity property is only true on a critical region of [0, 1]2 that depends on L∗ parameters. Simultaneous fractional convergence remains true on all of [0, 1]2 . We need   Definition 6.6 Let α1 , α2 > 0; α = (α1 , α2 ), f ∈ C [0, 1]2 , and let x = (x1 , x2 ), (t1 , t2 ) ∈ [0, 1]2 . We define the left mixed Riemann-Liouville fractional two dimensional integral of order α (see also [7]):  α  I0+ f (x) :=

1  (α1 )  (α2 )

x1 x2 0

(x1 − t1 )α1 −1 (x2 − t2 )α2 −1 f (t1 , t2 ) dt1 dt2 ,

0

(6.10) with x1 , x2 > 0. α Notice here that I0+ (|f |) < ∞. Definition 6.7 ([2]) Let α1 , α2 > 0 with  α1 = m1 , α2 = m2 , (· ceiling of the number). Let here f ∈ C m1 ,m2 [0, 1]2 . We consider the left (Caputo type) fractional partial derivative: (α1 ,α2 ) f (x) := D∗0

x1 x2 0

1 ·  (m1 − α1 )  (m2 − α2 )

(x1 − t1 )m1 −α1 −1 (x2 − t2 )m2 −α2 −1

0

∂ m1 +m2 f (t1 , t2 ) dt1 dt2 , ∂t1m1 ∂t2m2

(6.11)

∀ x = (x1 , x2 ) ∈ [0, 1]2 , where  is the gamma function

∞  (ν) =

e−t t ν−1 dt, ν > 0.

(6.12)

0

We set

(0,0) f (x) := f (x) , ∀ x ∈ [0, 1]2 ; D∗0 (m1 ,m2 ) f (x) := D∗0

∂ m1 +m2 f (x) , ∀ x ∈ [0, 1]2 . ∂x1m1 ∂x2m2

(6.13) (6.14)

6.1 Introduction

125

Definition 6.8 ([2]) We also set (0,α2 ) D∗0 f

(α1 ,0) D∗0 f

x2

1 (x) :=  (m2 − α2 )

(x2 − t2 )m2 −α2 −1

∂ m2 f (x1 , t2 ) dt2 , ∂t2m2

(6.15)

(x1 − t1 )m1 −α1 −1

∂ m1 f (t1 , x2 ) dt1 , ∂t1m1

(6.16)

(x2 − t2 )m2 −α2 −1

∂ m1 +m2 f (x1 , t2 ) dt2 , ∂x1m1 ∂t2m2

(6.17)

(x1 − t1 )m1 −α1 −1

∂ m1 +m2 f (t1 , x2 ) dt1 . ∂t1m1 ∂x2m2

(6.18)

0

x1

1 (x) :=  (m1 − α1 )

0

and (m1 ,α2 ) f D∗0

(α1 ,m2 ) D∗0 f

1 (x) :=  (m2 − α2 ) 1 (x) :=  (m1 − α1 )

x2 0

x1 0

The following result is the main motivation for this chapter: Theorem 6.9 ([2]) Let h1 , h2 , v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ C r,p [0, 1]2 . Let αij (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real valued functions, defined and bounded in [0, 1]2 and assume αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout [0, 1]2 . Let 0 ≤ α1h1 ≤ h1 < α11≤ h1+ 1 < α12 ≤ h1 + 2 < α13 ≤ h1 + 3 < ... < h1v1 ≤ v1 < ... < α1r ≤ r, with α1h1 = h1 ; 0 ≤ α2h2 ≤ h2 < α21 ≤ h2 + 1 < α22 ≤ h2 + 2 < α23 ≤ h2 + 3 < ... < α2v2 ≤ v2 < ... < α2p ≤ p, with α2h2 = h2 . Consider the left fractional differential bivariate operator v1  v2  (α1i ,α2j ) αij (x, y) D∗0 . (6.19) L∗ := i=h1 j=h2

Let integers m1 , m2 with m1 > r, m2 > p. Set lij :=

sup (x,y)∈[0,1]2

−1 α h1 h2 (x, y) · αij (x, y) < ∞.

  Also set (α1i = i, α2j = j, · ceiling of number) i,j

i,j

Mm1 ,m2 := Mm1 ,m2 (f ) :=

(6.20)

   1 1 1 (i,j)   t (i, j) ω1 f ; √ ,  (i − α1i + 1)  j − α2j + 1 m1 − i m2 − j

126

6 Bivariate Polynomial Abstract Left and Right …

+ max

  i (i − 1) j (j − 1) · f (i,j) ∞ , , m1 m2

i = h1 , ..., v1 ; j = h2 , ..., v2 . Here t is a positive real-valued function on Z2+ , ·∞ is the supremum norm on [0, 1]2 . Call v1  v2  i,j Pm1 ,m2 := Pm1 ,m2 (f ) = lij · Mm1 ,m2 . (6.21) i=h1 j=h2

Then there exists a polynomial Qm1 ,m2 (x, y) of degree (m1 , m2 ) on [0, 1]2 such that  (α1k ,α2l ) (α1k ,α2l )  Qm1 ,m2 (f ) − D∗0 D∗0

∞

≤

 (h1 − k + 1)  (h2 − l + 1) Pm1 ,m2 + Mmk,l1 ,m2 ,  (h1 − α1k + 1)  (h2 − α2l + 1) (h1 − k)! (h2 − l)!

(6.22)

for (0, 0) ≤ (k, l) ≤ (h1 , h2 ) . If (h1 + 1, h2 + 1) ≤ (k, l) ≤ (r, p), or 0 ≤ k ≤ h1 , h2 + 1 ≤ l ≤ p, or h1 + 1 ≤ k ≤ r, 0 ≤ l ≤ h2 , we get  (α1k ,α2l ) (α1k ,α2l )  Qm1 ,m2 (f ) − D∗0 D∗0

∞

≤ Mmk,l1 ,m2 .

(6.23)

  By assuming L∗ (f (1, 1)) ≥ 0, we get L∗ Qm1 ,m2 (1, 1) ≥ 0. Let 1 ≥ x, y > 0, with α1h1 = h1 and α2h2 = h2 , such that    1 x ≥  h1 − α1h1 + 1 (h1 −α1h1 ) ,

(6.24)

   1 y ≥  h2 − α2h2 + 1 (h2 −α2h2 ) , and

Then

L∗ (f (x, y)) ≥ 0.   L∗ Qm1 ,m2 (x, y) ≥ 0.

Some notation follows: Definition 6.10 Let f be a real-valued function defined on [0, 1]2 and let m1 , m2 ∈ N. Let Bm1 ,m2 be the Bernstein (polynomial) operator of order (m1 , m2 ) given by Bm1 ,m2 (f ; x1 , x2 ) :=

6.1 Introduction

127

    m1  m2  i1 i2 m1 m2 x1i1 (1 − x1 )m1 −i1 x2i2 (1 − x2 )m2 −i2 . f , i1 i2 m m 1 2 i =0 i =0 1

(6.25)

2

In this chapter we generalize Theorem 6.9 to abstract kernels that can be singular or non-singular, again bivariate constrained monotonicity takes place over a critical region of [0, 1]2 , however bivariate abstract fractional simultaneous approximation is true over the whole of [0, 1]2 . We cover both the left and right sides of this bivariate fractional approximation. We need the following abstract fractional background.

6.2 About Bivariate Abstract Fractional Calculus Let h1 , h2 , v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ 2 / Z+ : α1i = C r,p  [0,  1] . Here h1 ≤ i ≤ v1 , h2 ≤ j ≤ v2 . Let α1i , α2j ≥ 0, α1i , α2j ∈ i, α2j = j, i = 0, 1, ..., r; j = 0, 1, ..., p, (· is the ceiling of number), α10 = 0, α20 = 0. Consider also the integrable functions k1i := Kα1i , k2j := Kα2j : [0, 1] → R+ , i = 0, 1, ..., r; j = 0, 1, ..., p. (I) We first considerthe abstract left Caputo type bivariate fractional partial derivative of orders α1i , α2j k1i (α1i ,α2j ) f k2j D∗0

x1 x2 (x) :=

k1i (x1 − t1 ) k2j (x2 − t2 ) 0

0

∀ x = (x1 , x2 ) ∈ [0, 1]2 . We set k1i (i,j) k2j D∗0 f

k10 (0,0) k20 D∗0 f

(x) :=

k1i (i,α2j ) f k2j D∗0

k2j (x2 − t2 )

x1

(x) :=

k1i (x1 − t1 ) 0

(6.26)

(6.27)

0

k1i (α1i ,j) f k2j D∗0

dt1 dt2 ,

∀ x = (x1 , x2 ) ∈ [0, 1]2 .

x2 (x) :=

j

∂t1i ∂t2

(x) := f (x) ,

∂ i+j f (x1 ,x2 ) , j ∂x1i ∂x2

We also set

∂ i+j f (t1 , t2 )

∂ i+j f (x1 , t2 ) j

∂x1i ∂t2 ∂ i+j f (t1 , x2 ) j

∂t1i ∂x2

dt2 ,

(6.28)

dt1 ,

(6.29)

128

6 Bivariate Polynomial Abstract Left and Right …

and in particular we define: k10 (0,α2j ) f k2j D∗0

x2 (x) :=

k2j (x2 − t2 ) 0

k1i (α1i ,0) f k20 D∗0

x1 (x) :=

k1i (x1 − t1 ) 0

∂ j f (x1 , t2 )

dt2 ,

(6.30)

∂ i f (t1 , x2 ) dt1 , ∂t1i

(6.31)

j

∂t2

∀ x = (x1 , x2 ) ∈ [0, 1]2 . We will assume that there exists a critical region ∅ = ⊆ [0, 1]2 such that :=

⎧ ⎨ ⎩

x (x, y) ∈ [0, 1]2 :

⎫ ⎬

y

k2h2 (z) dz ≥ 1, for any of h1 , h2 = 0 . ⎭

k1h1 (z) dz, 0

0

(6.32)     In [2], we got that 0 <  h1 − α1h1 + 1 ,  h2 − α2h2 + 1 ≤ 1, where  is the gamma function, and there it is kihi (z) = and it holds

z hi −αihi −1   , i = 1, 2; ∀ z ∈ [0, 1] ,  hi − αihi

1 kihi (z) dz = 0

(6.33)

1   ≥ 1,  hi − αihi + 1

(6.34)

proving there that = ∅, by x = y = 1. Also in [2], see Theorem 6.9, when α1h1 = h1 and α2h2 = h2 and 0 < x, y < 1 the critical region there contains all (x, y) such that ⎫  h −α1   ⎬ ( 1 1h1 ) , 1 > x ≥  h1 − α1h1 + 1 , 1    ⎭ ⎩ 1 > y ≥  h2 − α2h2 + 1 (h2 −α2h2 ) . ⎧ ⎨

(6.35)

so again = ∅, non-trivially. (II) We also consider right side Caputo type bivariate fractional partial   the abstract derivative of orders α1i , α2j k1i (α1i ,α2j ) f k2j D1−

1 1 (x) := (−1)i+j

k1i (t1 − x1 ) k2j (t2 − x2 ) x1 x2

∂ i+j f (t1 , t2 ) j

∂t1i ∂t2

dt1 dt2 , (6.36)

∀ x = (x1 , x2 ) ∈ [0, 1]2 .

6.2 About Bivariate Abstract Fractional Calculus

129

We set k10 (0,0) k20 D1− f k1i (i,j) k2j D1− f

(x) := (−1)i+j

(x) := f (x) ,

∂ i+j f (x1 ,x2 ) , j ∂x1i ∂x2

(6.37) ∀ x = (x1 , x2 ) ∈ [0, 1]2 .

We also set k1i (i,α2j ) f k2j D1−

1 (x) := (−1)

k2j (t2 − x2 )

j

∂ i+j f (x1 , t2 )

k1i (α1i ,j) k2j D1− f

1 (x) := (−1)

k1i (t1 − x1 )

i

dt2 ,

(6.38)

dt1 ,

(6.39)

dt2 ,

(6.40)

∂ i f (t1 , x2 ) dt1 , ∂t1i

(6.41)

j

∂x1i ∂t2

x2

∂ i+j f (t1 , x2 )

x1

j

∂t1i ∂x2

and in particular we define: k10 (0,α2j ) f k2j D1−

1 (x) := (−1)

k2j (t2 − x2 )

j x2

k1i (α1i ,0) f k20 D1−

1 (x) := (−1)i

k1i (t1 − x1 ) x1

∂ j f (x1 , t2 ) j

∂t2

∀ x = (x1 , x2 ) ∈ [0, 1]2 . We will assume that there exists a critical region ∅ = ⊆ [0, 1]2 such that ⎧ ⎨

⎫

1−x

1−y ⎬

:= (x, y) ∈ [0, 1]2 : k1h1 (z) dz, k2h2 (z) dz ≥ 1, for any of h1 , h2 = 0 . ⎩ ⎭ 0

0

(6.42) When x = y = 0 we have the conditions in (6.42) fulfilled by (6.33), (6.34), so there

= ∅, see [3]. Also in [3], when α1h1 = h1 and α2h2 = h2 and 0 < x, y < 1 the critical region

there contains all (x, y) such that ⎧ ⎨

 1   1 − x ≥  h1 − α1h1 + 1 (h1 −α1h1 )    1 ⎩ 1 − y ≥  h2 − α2h2 + 1 (h2 −α2h2 )

, .

⎫ ⎬ ⎭

,

(6.43)

130

6 Bivariate Polynomial Abstract Left and Right …

equivalently, ⎧ ⎨

 1   0 < x ≤ 1 −  h1 − α1h1 + 1 (h1 −α1h1 )    1 ⎩ 0 < y ≤ 1 −  h2 − α2h2 + 1 (h2 −α2h2 )

, .

⎫ ⎬ ⎭

,

(6.44)

so again = ∅, non-trivially.

6.3 Main Results We present our left side result: Theorem 6.11 Let  h1 , h2, v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ C r,p [0, 1]2 . Let αij (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real valued functions, defined and bounded in [0, 1]2 . We follow the terminology of section 6.2. We assume that αh1 h2 is either ≥ α > 0 or ≤ β < 0 throughout  the  critical / Z+ : α1i = i, α2j = j, i = region , see (6.32). Let also α1i , α2j ≥ 0, α1i , α2j ∈ 0, 1, ..., r; j = 0, 1, ..., p, α10 = 0, α20 = 0. Consider the bivariate left fractional differential operator v1  v2  (α1i ,α2j ) αij (x, y) D∗0 . (6.45) L∗ := i=h1 j=h2

Let the integers m1 , m2 with m1 > r, m2 > p. Set lij := αh−1 (x, y) αij (x, y) ∞,[0,1]2 < ∞. 1 h2

(6.46)

Also set  i,j Mm1 ,m2

:=

i,j Mm1 ,m2



(f ) := λ1i λ2j t (i, j) ω1 f

(i,j)

1 1 ;√ , m1 − i m2 − j

  i (i − 1) j (j − 1) (i,j) max , f∞,[0,1]2 , m1 m2

 +



(6.47)

for i = 0, 1, ..., r; j = 0, 1, ..., p. Above it is t : Z2+ → R+ , and

1 λ1i :=

1 k1i (z) dz, λ2j :=

0

for i = 1, ..., r; j = 1, ..., p, and λ10 := λ20 := 1.

k2j (z) dz 0

(6.48)

6.3 Main Results

131

Call Pm1 ,m2 := Pm1 ,m2 (f ) :=

v1  v2 

i,j

lij Mm1 ,m2 .

(6.49)

i=h1 j=h2

Then there exists a polynomial Qm1 ,m2 (x, y) of degree (m1 , m2 ) on [0, 1]2 such that (i)  k1i (α1i ,α2j ) (α1i ,α2j )  Qm1 ,m2 ≤ (f ) − kk1i2j D∗0 k2j D∗0 2 ∞,[0,1]

Pm1 ,m2 k1i (α1i ,α2j )  h1 h2  i,j x y + Mm1 ,m2 , k2j D∗0 ∞,[0,1]2 h1 !h2 !

(6.50)

for (0, 0) ≤ (i, j) ≤ (h1 , h2 ) , and (ii) if (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 , k1i (α1i ,α2j ) (f ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Qm1 ,m2

i,j

∞,[0,1]2

≤ Mm1 ,m2 .

(6.51)

  By assuming L∗ f (x, y) ≥ 0, ∀ (x, y) ∈ , we obtain L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈ . Proof We observe that (m1 , m2 ∈ N : m1 > i, m2 > j): k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Bm1 ,m2 f (x1 , x2 ) =

x x 1 2 ∂ i+j f (t1 , t2 ) k1i (x1 − t1 ) k2j (x2 − t2 ) dt1 dt2 − j ∂t1i ∂t2 0

0

x1 x2 k1i (x1 − t1 ) k2j (x2 − t2 ) 0

  ∂ i+j Bm1 ,m2 f (t1 , t2 ) j

∂t1i ∂t2

0

dt1 dt2 =

(6.52)

x x     1 2 i+j i+j B f , t ∂ (t ) ∂ f , t (t ) m1 ,m2 1 2 1 2 k1i (x1 − t1 ) k2j (x2 − t2 ) − dt1 dt2 ≤ i j i j ∂t1 ∂t2 ∂t1 ∂t2 0

0

x1 x2 0

0

 ∂ i+j f (t , t ) ∂ i+j B (6.3) m1 ,m2 f (t1 , t2 ) 1 2 dt1 dt2 ≤ k1i (x1 − t1 ) k2j (x2 − t2 ) − j j ∂t i ∂t ∂t i ∂t 1

2

1

2

132

6 Bivariate Polynomial Abstract Left and Right …

⎛ ⎝

x1 x2 0





t (i, j) ω1

⎞ k1i (x1 − t1 ) k2j (x2 − t2 ) dt1 dt2 ⎠

0

1 1 f (i,j) ; √ , m1 − i m2 − j

⎛ ⎝

x1



  i (i − 1) j (j − 1) (i,j) + max , = f ∞,[0,1]2 m1 m2

⎞⎛ x ⎞

2 k1i (x1 − t1 ) dt1 ⎠ ⎝ k2j (x2 − t2 ) dt2 ⎠

0





t (i, j) ω1

0

1 1 , f (i,j) ; √ m1 − i m2 − j

⎛ ⎝

x1



  i (i − 1) j (j − 1) (i,j) + max , = f ∞,[0,1]2 m1 m2

⎞⎛ x ⎞

2 k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠

0





t (i, j) ω1

(6.53)

1 1 , f (i,j) ; √ m1 − i m2 − j

(6.54)

0



  i (i − 1) j (j − 1) (i,j) + max , ≤ f ∞,[0,1]2 m1 m2

⎛ 1 ⎞⎛ 1 ⎞

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0



 t (i, j) ω1

1 1 , f (i,j) ; √ m1 − i m2 − j

0



  i (i − 1) j (j − 1) (i,j) + max , . f ∞,[0,1]2 m1 m2

We have proved that k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Bm1 ,m2 f (x1 , x2 ) ≤

(6.55)

⎛ 1 ⎞⎛ 1 ⎞

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

 t (i, j) ω1



1 1 , f (i,j) ; √ m1 − i m2 − j

0



  i (i − 1) j (j − 1) (i,j) + max , , f ∞,[0,1]2 m1 m2

∀ (x1 , x2 ) ∈ [0, 1]2 ; i = 1, ..., r; j = 1, ..., p; where m1 , m2 ∈ N : m1 > r, m2 > p.

6.3 Main Results

133

So we have established that k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Bm1 ,m2 f (x1 , x2 )

∞,[0,1]2

≤

(6.56)

⎞⎛ 1 ⎞ ⎛ 1

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0





t (i, j) ω1

0

1 1 , f (i,j) ; √ m1 − i m2 − j



  i (i − 1) j (j − 1) (i,j) + max , , f ∞,[0,1]2 m1 m2

for i = 1, ..., r; j = 1, ..., p; m1 , m2 ∈ N : m1 > r, m2 > p. We call

1

1 λ1i := k1i (z) dz, λ2j := k2j (z) dz, 0

(6.57)

0

for i = 1, ..., r; j = 1, ..., p, as in (6.48). We also set λ10 := λ20 := 1. Thus, the following inequality is valid in general: k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D∗0 

 t (i, j) ω1

k1i (α1i ,α2j ) k2j D∗0

1 1 , f (i,j) ; √ m1 − i m2 − j



  Bm1 ,m2 f (x1 , x2 )

∞,[0,1]2

≤ λ1i λ2j

  i (i − 1) j (j − 1) (i,j) + max , f ∞,[0,1]2 m1 m2

i,j

= Mm1 ,m2 ,

(6.58)

for i = 0, 1, ..., r; j = 0, 1, ..., p; m1 , m2 ∈ N : m1 > r, m2 > p, case of Theorem 6.4. Case (i) Assume throughout that αh1 h2 (x, y) ≥ α > 0. Call Qm1 ,m2 (x, y) := Bm1 ,m2 (f ; x, y) + Pm1 ,m2

xh1 yh2 , h1 ! h2 !

(6.59)

∀ (x, y) ∈ [0, 1]2 . To remind, here it is lij := αh−1 (x, y) αij (x, y) ∞,[0,1]2 , 1 h2 and Pm1 ,m2 :=

v1  v2  i=h1 j=h2

i,j

lij Mm1 ,m2 .

(6.60)

134

6 Bivariate Polynomial Abstract Left and Right …

Therefore by (6.58) we obtain   k1i (α1i ,α2j ) xh1 yh2 D f + Pm1 ,m2 − k2j ∗0 h !h ! 1

2

k1i (α1i ,α2j ) k2j D∗0

  Qm1 ,m2

i,j

∞,[0,1]2

≤ Mm1 ,m2 , (6.61)

all 0 ≤ i ≤ r, 0 ≤ j ≤ p. Let (0, 0) ≤ (i, j) ≤ (h1 , h2 ), by (6.61) we get that k1i (α1i ,α2j ) (f ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Qm1 ,m2

∞,[0,1]2

≤

Pm1 ,m2 k1i (α1i ,α2j )  h1 h2  i,j x y + Mm1 ,m2 , k2j D∗0 ∞,[0,1]2 h1 !h2 !

(6.62)

proving (6.50). If (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 we get by (6.61) that k1i (α1i ,α2j ) (f ) − k2j D∗0

k1i (α1i ,α2j ) k2j D∗0

  Qm1 ,m2

i,j

∞,[0,1]2

≤ Mm1 ,m2 ,

(6.63)

proving (6.51). For (x, y) in the critical region , we can write   αh−1 (x, y) L∗ Qm1 ,m2 (x, y) = αh−1 (x, y) L∗ (f (x, y)) + 1 h2 1 h2 Pm1 ,m2 h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D∗0

v1  v2  h1 h2 !  αh−1 x y + (x, y) αij (x, y) 1 h2

k1i (α1i ,α2j ) k2j D∗0

i=h1 j=h2

  xh1 yh2 (6.61) Qm1 ,m2 (x, y) − f (x, y) − Pm1 ,m2 ≥ h1 ! h2 ! (by L∗ f ≥ 0) Pm1 ,m2 h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D∗0

v1  v2  h1 h2 !  i,j x y − lij Mm1 ,m2 = i=h1 j=h2

⎡k Pm1 ,m2 ⎣

(α1h1 ,α2h2 ) 1h1 k2h2 D∗0 h1 !h2 !

 h h x 1y 2

⎤ − 1⎦ =

(6.64)

6.3 Main Results

135

⎡ ⎢ Pm1 ,m2 ⎣

k1h1 (α1h1 ,0) h x1 k20 D∗0

!

(0,α2h2 ) h2 k10 y k2h2 D∗0

⎤

!

⎥ − 1⎦ =: ϕ.

h1 !h2 !

(6.65)

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ϕ = 0. If h1 = 0, and h2 = 0, then ⎡ ϕ = Pm1 ,m2 ⎣ ⎡ Pm1 ,m2 ⎣

y

(0,α2h2 ) h2 k10 y k2h2 D∗0 h2 !

⎤ − 1⎦ =

⎛ y ⎞

k2h2 (y − t) dt − 1⎦ = Pm1 ,m2 ⎝ k2h2 (z) dz − 1⎠ ≥ 0. ⎤

0

(6.66)

0

Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡⎛ x ⎞⎛ y ⎞ ⎤

ϕ = Pm1 ,m2 ⎣⎝ k1h1 (x − t) dt ⎠ ⎝ k2h2 (y − t) dt ⎠ − 1⎦ = 0

0

⎡⎛ x ⎞⎛ y ⎞ ⎤

Pm1 ,m2 ⎣⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠ − 1⎦ ≥ 0. 0

(6.67)

0

So in all four cases we have proved that   L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈ .

(6.68)

Case (ii) Assume throughout that αh1 h2 ≤ β < 0. Consider Qm1 ,m2 (x, y) := Bm1 ,m2 (f ; x, y) − Pm1 ,m2

xh1 yh2 , h1 ! h2 !

∀ (x, y) ∈ [0, 1]2 . Therefore by (6.58) we obtain   k1i (α1i ,α2j ) xh1 yh2 D f − P − m1 ,m2 k2j ∗0 h !h ! 1

2

k1i (α1i ,α2j ) k2j D∗0

  Qm1 ,m2

i,j

∞,[0,1]

2

≤ Mm1 ,m2 ,

(6.69) all 0 ≤ i ≤ r, 0 ≤ j ≤ p. Also both (6.50), (6.51) are valid, just replace Qm1 ,m2 by Qm1 ,m2 in (6.62), (6.63). For (x, y) in the critical region , we can write

136

6 Bivariate Polynomial Abstract Left and Right …

  αh−1 (x, y) L∗ Qm1 ,m2 (x, y) = αh−1 (x, y) L∗ (f (x, y)) − 1 h2 1 h2 k1h1 (α1h1 ,α2h2 ) k2h2 D∗0

Pm1 ,m2 h1 !h2 !

v1  v2  h1 h2 !  x y + αh−1 (x, y) αij (x, y) 1 h2

k1i (α1i ,α2j ) k2j D∗0

i=h1 j=h2

  xh1 yh2 (6.69) Qm1 ,m2 (x, y) − f (x, y) + Pm1 ,m2 ≤ h1 ! h2 !

(6.70)

(by L∗ f ≥ 0) −

k1h1 (α1h1 ,α2h2 ) k2h2 D∗0

Pm1 ,m2 h1 !h2 !

v1  v2  h1 h2 !  i,j x y + lij Mm1 ,m2 = i=h1 j=h2

⎡ Pm1 ,m2 ⎣1 − ⎡ ⎢ Pm1 ,m2 ⎣1 −

k1h1 (α1h1 ,α2h2 ) k2h2 D∗0

h1 !h2 !

k1h1 (α1h1 ,0) h x1 k20 D∗0

!

 h h ⎤ x 1y 2 ⎦=

(0,α2h2 ) h2 k10 y k2h2 D∗0

!⎤

h1 !h2 !

⎥ ⎦ =: ψ.

(6.71)

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ψ = 0. If h1 = 0, and h2 = 0, then ⎡ ϕ = Pm1 ,m2 ⎣1 − ⎡ Pm1 ,m2 ⎣1 −

y

(0,α2h2 ) h2 k10 y k2h2 D∗0

⎤

h2 ! ⎡

k2h2 (y − t) dt ⎦ = Pm1 ,m2 ⎣1 −

0

y

⎤ ⎦= ⎤ k2h2 (z) dz ⎦ ≤ 0.

(6.72)

0

Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡

⎛

ϕ = Pm1 ,m2 ⎣1 − ⎝

x

⎞⎛ k1h1 (x − t) dt ⎠ ⎝

0

⎡

y

⎞⎤ k2h2 (y − t) dt ⎠⎦ =

0

⎛

⎞⎛ y ⎞⎤

x

Pm1 ,m2 ⎣1 − ⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠⎦ ≤ 0. 0

0

(6.73)

6.3 Main Results

137

So in all four cases we have proved that   L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈ .

(6.74) 

Next we give our right side result: Theorem 6.12 Let h1 , h2 be even  and v1 , v2 , r, p be integers, 0 ≤ h1 ≤ v1 ≤ r, 0 ≤ h2 ≤ v2 ≤ p and let f ∈ C r,p [0, 1]2 . Let αij (x, y), i = h1 , h1 + 1, ..., v1 ; j = h2 , h2 + 1, ..., v2 be real valued functions, defined and bounded in [0, 1]2 . We follow the terminology of section 6.2. We assume that αh1 h2 is either ≥ α > 0 or ≤ β < 0 / Z+ : throughoutthecritical region , see (6.42). Let also α1i , α2j ≥ 0, α1i , α2j ∈ α1i = i, α2j = j, i = 0, 1, ..., r; j = 0, 1, ..., p, α10 = 0, α20 = 0. Consider the bivariate right fractional differential operator ∗

L :=

v1  v2 

(α1i ,α2j ) αij (x, y) D1− .

(6.75)

i=h1 j=h2

Let the integers m1 , m2 with m1 > r, m2 > p. Set lij := αh−1 (x, y) αij (x, y) ∞,[0,1]2 < ∞. 1 h2

(6.76)

Also set  i,j Mm1 ,m2

:=

i,j Mm1 ,m2



(f ) := λ1i λ2j t (i, j) ω1 f

max

(i,j)

1 1 ;√ , m1 − i m2 − j

  i (i − 1) j (j − 1) (i,j) , f∞,[0,1]2 , m1 m2

 +

(6.77)

for i = 0, 1, ..., r; j = 0, 1, ..., p. Above it is t : Z2+ → R+ , and

1 λ1i :=

1 k1i (z) dz, λ2j :=

0

k2j (z) dz

(6.78)

0

for i = 1, ..., r; j = 1, ..., p, and λ10 := λ20 := 1. Call v1  v2  i,j Pm1 ,m2 := Pm1 ,m2 (f ) := lij Mm1 ,m2 .

(6.79)

i=h1 j=h2

Then there exists a polynomial Qm1 ,m2 (x, y) of degree (m1 , m2 ) on [0, 1]2 such that

138

6 Bivariate Polynomial Abstract Left and Right …

(i)

k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

∞,[0,1]2

≤

Pm1 ,m2 k1i (α1i ,α2j )  h1 h2  i,j x y + Mm1 ,m2 , k2j D1− ∞,[0,1]2 h1 !h2 !

(6.80)

for (0, 0) ≤ (i, j) ≤ (h1 , h2 ) , and (ii) if (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 , k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

i,j

∞,[0,1]2

≤ Mm1 ,m2 .

(6.81)

  By assuming L∗ f (x, y) ≥ 0, ∀ (x, y) ∈ , we obtain L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈

. Proof We observe that (m1 , m2 ∈ N : m1 > i, m2 > j): k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Bm1 ,m2 f (x1 , x2 ) =



1 1 ∂ i+j f (t1 , t2 ) (−1)i+j k − x − x dt1 dt2 − k (t ) (t ) 1i 1 1 2j 2 2 j ∂t1i ∂t2 x1 x2

1 1 k1i (t1 − x1 ) k2j (t2 − x2 )

(−1)i+j

∂

i+j

  Bm1 ,m2 f (t1 , t2 ) j

∂t1i ∂t2

x1 x2

dt1 dt2 =

(6.82)

1 1     i+j i+j f , t B ∂ (t ) ∂ f (t1 , t2 ) m1 ,m2 1 2 k − x − x − dt dt k (t ) (t ) 1i 1 1 2j 2 2 1 2 ≤ j j ∂t1i ∂t2 ∂t1i ∂t2 x1 x2

1 1 x1 x2

 ∂ i+j f (t , t ) ∂ i+j B (6.3) m1 ,m2 f (t1 , t2 ) 1 2 k1i (t1 − x1 ) k2j (t2 − x2 ) − dt1 dt2 ≤ j j i i ∂t ∂t ∂t ∂t 1

⎛ ⎝

1 1

2

1

2

⎞ k1i (t1 − x1 ) k2j (t2 − x2 ) dt1 dt2 ⎠

(6.83)

x1 x2

 t (i, j) ω1



1 1 , f (i,j) ; √ m1 − i m2 − j



  i (i − 1) j (j − 1) (i,j) , + max = f ∞,[0,1]2 m1 m2

6.3 Main Results

139

⎛ ⎝

1

⎞⎛ 1 ⎞

k1i (t1 − x1 ) dt1 ⎠ ⎝ k2j (t2 − x2 ) dt2 ⎠

x1





t (i, j) ω1 f (i,j) ; √

x2

1 1 , m1 − i m2 − j



+ max

  i (i − 1) j (j − 1) (i,j) , = f ∞,[0,1]2 m1 m2

⎛ 1−x ⎞ ⎛ 1−x ⎞

1

2 ⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0





t (i, j) ω1

1 1 , f (i,j) ; √ m1 − i m2 − j

(6.84)

0



  i (i − 1) j (j − 1) (i,j) + max , ≤ f ∞,[0,1]2 m1 m2

⎛ 1 ⎞⎛ 1 ⎞

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0



 t (i, j) ω1

1 1 , f (i,j) ; √ m1 − i m2 − j

0



  i (i − 1) j (j − 1) (i,j) , + max . f ∞,[0,1]2 m1 m2

We have proved that k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Bm1 ,m2 f (x1 , x2 ) ≤

(6.85)

⎛ 1 ⎞⎛ 1 ⎞

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0





t (i, j) ω1 f (i,j) ; √

1 1 , m1 − i m2 − j

0



+ max

  i (i − 1) j (j − 1) (i,j) , , f ∞,[0,1]2 m1 m2

∀ (x1 , x2 ) ∈ [0, 1]2 ; i = 1, ..., r; j = 1, ..., p; where m1 , m2 ∈ N : m1 > r, m2 > p. So we have established that  k1i (α1i ,α2j ) (α1i ,α2j )  Bm1 ,m2 f (x1 , x2 ) f (x1 , x2 ) − kk1i2j D1− ≤ (6.86) k2j D1− 2 ∞,[0,1]

⎛ 1 ⎞⎛ 1 ⎞

⎝ k1i (z) dz ⎠ ⎝ k2j (z) dz ⎠ 0

0

140

6 Bivariate Polynomial Abstract Left and Right …





t (i, j) ω1

1 1 , f (i,j) ; √ m1 − i m2 − j



  i (i − 1) j (j − 1) (i,j) , + max , f ∞,[0,1]2 m1 m2

for i = 1, ..., r; j = 1, ..., p; m1 , m2 ∈ N : m1 > r, m2 > p. We call

1

1 λ1i := k1i (z) dz, λ2j := k2j (z) dz, 0

(6.87)

0

for i = 1, ..., r; j = 1, ..., p, as in (6.78). We also set λ10 := λ20 := 1. Thus, the following inequality is valid in general: k1i (α1i ,α2j ) f (x1 , x2 ) − k2j D1− 

 t (i, j) ω1

k1i (α1i ,α2j ) k2j D1−

1 1 , f (i,j) ; √ m1 − i m2 − j



  Bm1 ,m2 f (x1 , x2 )

≤ λ1i λ2j

∞,[0,1]2

  i (i − 1) j (j − 1) (i,j) , + max f ∞,[0,1]2 m1 m2

i,j

= Mm1 ,m2 ,

(6.88)

for i = 0, 1, ..., r; j = 0, 1, ..., p; m1 , m2 ∈ N : m1 > r, m2 > p, case of Theorem 6.4. Case (i) Assume throughout that αh1 h2 (x, y) ≥ α > 0. Call Qm1 ,m2 (x, y) := Bm1 ,m2 (f ; x, y) + Pm1 ,m2

xh1 yh2 , h1 ! h2 !

(6.89)

∀ (x, y) ∈ [0, 1]2 . To remind, here it is , y) α y) lij := αh−1 (x, (x, ij h 1 2 ∞,[0,1]2 and Pm1 ,m2 :=

v1  v2 

(6.90)

i,j

lij Mm1 ,m2 .

i=h1 j=h2

Therefore by (6.88) we obtain   k1i (α1i ,α2j ) xh1 yh2 D f + Pm1 ,m2 − k2j 1− h !h ! 1

all 0 ≤ i ≤ r, 0 ≤ j ≤ p.

2

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

i,j

∞,[0,1]

2

≤ Mm1 ,m2 , (6.91)

6.3 Main Results

141

Let (0, 0) ≤ (i, j) ≤ (h1 , h2 ), by (6.91) we get that k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

∞,[0,1]2

≤

Pm1 ,m2 k1i (α1i ,α2j )  h1 h2  i,j x y + Mm1 ,m2 , k2j D1− ∞,[0,1]2 h1 !h2 !

(6.92)

proving (6.80). If (h1 + 1, h2 + 1) ≤ (i, j) ≤ (r, p), or 0 ≤ i ≤ h1 , h2 + 1 ≤ j ≤ p, or h1 + 1 ≤ i ≤ r, 0 ≤ j ≤ h2 we get by (6.91) that k1i (α1i ,α2j ) (f ) − k2j D1−

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

i,j

∞,[0,1]2

≤ Mm1 ,m2 ,

(6.93)

proving (6.81). For (x, y) in the critical region , we can write   αh−1 (x, y) L∗ Qm1 ,m2 (x, y) = αh−1 (x, y) L∗ (f (x, y)) + 1 h2 1 h2 Pm1 ,m2 h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

v1  v2  h1 h2 !  αh−1 x y + (x, y) αij (x, y) 1 h2

k1i (α1i ,α2j ) k2j D1−

(6.94)

i=h1 j=h2

  xh1 yh2 (6.91) Qm1 ,m2 (x, y) − f (x, y) − Pm1 ,m2 ≥ h1 ! h2 ! (by L∗ f ≥ 0) k1h1 (α1h1 ,α2h2 ) k2h2 D1−

Pm1 ,m2 h1 !h2 !

v1  v2  h1 h2 !  i,j x y − lij Mm1 ,m2 = i=h1 j=h2

⎡k

1h1

Pm1 ,m2 ⎣ ⎡ ⎢ Pm1 ,m2 ⎣

k2h2

 (α1h ,α2h )  D1− 1 2 xh1 yh2 h1 !h2 !

k1h1 (α1h1 ,0) h x1 k20 D1−

!

⎤ − 1⎦ =

(0,α2h2 ) h2 k10 y k2h2 D1−

h1 !h2 !

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ψ = 0.

!

⎤ ⎥ − 1⎦ =: ψ.

(6.95)

142

6 Bivariate Polynomial Abstract Left and Right …

If h1 = 0, and h2 = 0, then ⎡ ψ = Pm1 ,m2 ⎣ ⎡ Pm1 ,m2 ⎣(−1)h2

1

(0,α2h2 ) h2 k10 y k2h2 D1− h2 !

⎤ − 1⎦ = ⎤

(h is even) k2h2 (t − y) dt − 1⎦ 2 =

y

⎛ 1−y ⎞

Pm1 ,m2 ⎝ k2h2 (z) dz − 1⎠ ≥ 0.

(6.96)

0

Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡⎛ ψ = Pm1 ,m2 ⎣⎝

1

⎞⎛ k1h1 (t − x) dt ⎠ ⎝

x

1

⎞

⎤

k2h2 (t − y) dt ⎠ − 1⎦ =

y

⎡⎛ 1−x ⎞ ⎛ 1−y ⎞ ⎤

Pm1 ,m2 ⎣⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠ − 1⎦ ≥ 0. 0

(6.97)

0

So in all four cases we have proved that   L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈ .

(6.98)

Case (ii) Assume throughout that αh1 h2 ≤ β < 0. Consider Qm1 ,m2 (x, y) := Bm1 ,m2 (f ; x, y) − Pm1 ,m2

xh1 yh2 , h1 ! h2 !

∀ (x, y) ∈ [0, 1]2 . Therefore by (6.88) we obtain   k1i (α1i ,α2j ) xh1 yh2 D − f − Pm1 ,m2 k2j 1− h !h ! 1

2

k1i (α1i ,α2j ) k2j D1−

  Qm1 ,m2

i,j

∞,[0,1]

2

≤ Mm1 ,m2 ,

(6.99) all 0 ≤ i ≤ r, 0 ≤ j ≤ p. Also both (6.80), (6.81) are valid, just replace Qm1 ,m2 by Qm1 ,m2 in (6.92), (6.93). For (x, y) in the critical region , we can write

6.3 Main Results

143

  αh−1 (x, y) L∗ Qm1 ,m2 (x, y) = αh−1 (x, y) L∗ (f (x, y)) − 1 h2 1 h2 Pm1 ,m2 h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

v1  v2  h1 h2 !  x y + αh−1 (x, y) αij (x, y) 1 h2

k1i (α1i ,α2j ) k2j D1−

i=h1 j=h2

  xh1 yh2 (6.99) Qm1 ,m2 (x, y) − f (x, y) + Pm1 ,m2 ≤ h1 ! h2 !

(6.100)

(by L∗ f ≥ 0) −

Pm1 ,m2 h1 !h2 !

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

i=h1 j=h2

⎡ Pm1 ,m2 ⎣1 − ⎡ ⎢ Pm1 ,m2 ⎣1 −

v1  v2  h1 h2 !  i,j x y + lij Mm1 ,m2 =

k1h1 (α1h1 ,α2h2 ) k2h2 D1−

h1 !h2 !

k1h1 (α1h1 ,0) h x1 k20 D1−

!

 h h ⎤ x 1y 2 ⎦=

(0,α2h2 ) h2 k10 y k2h2 D1−

!⎤ ⎥ ⎦ =: ψ.

h1 !h2 !

(6.101)

If h1 = h2 = 0, then α1h1 = α2h2 = 0, and ψ = 0. If h1 = 0, and h2 = 0, then ⎡ ψ = Pm1 ,m2 ⎣1 − ⎡ Pm1 ,m2 ⎣1 −

1

(0,α2h2 ) h2 k10 y k2h2 D1− h2 !

⎤ ⎦=

⎡

⎤

⎤

1−y k2h2 (t − y) dt ⎦ = Pm1 ,m2 ⎣1 − k2h2 (z) dz ⎦ ≤ 0.

y

(6.102)

0

Similarly, we treat the case h1 = 0, h2 = 0. When h1 , h2 = 0, then we have ⎡

⎛

ψ = Pm1 ,m2 ⎣1 − ⎝

1

⎞⎛ k1h1 (t − x) dt ⎠ ⎝

x

1

⎞⎤ k2h2 (t − y) dt ⎠⎦ =

y

⎡

⎛ 1−x ⎞ ⎛ 1−y ⎞⎤

Pm1 ,m2 ⎣1 − ⎝ k1h1 (z) dz ⎠ ⎝ k2h2 (z) dz ⎠⎦ ≤ 0. 0

0

(6.103)

144

6 Bivariate Polynomial Abstract Left and Right …

So in all four cases we have proved that   L∗ Qm1 ,m2 (x, y) ≥ 0, ∀ (x, y) ∈ .

(6.104) 

Conclusion 6.13 Theorem 6.11 generalizes greatly Theorems 6.9, and 6.12 generalizes greatly the main Theorem 2.9 of [3]. These generalizations involve abstract fractional kernels that can be singular or non-singular kernels. That is we cover all kinds of left and right Caputo type fractional calculi of singular and non-singular kernel.

References 1. Anastassiou, G.A.: Bivariate monotone approximation. Proc. Amer. Math. Soc. 112(4), 959– 964 (1991) 2. Anastassiou, G.A.: Bivariate left fractional polynomial monotone approximation. In: Intelligent Mathematics II: Applied Mathematics and Approximation Theory, pp. 1–13, AMAT, Ankara, May 2015. Springer, New York (2016) 3. Anastassiou, G.A.: Bivariate right fractional polynomial monotone approximation. In: Computational Analysis, pp. 19–31, AMAT, Ankara, May 2015. Springer, New York (2016) 4. Anastassiou, G.A.: Bivariate abstract fractional monotone constrained approximation by polynomials (2021) (submitted) 5. Anastassiou, G.A., Shisha, O.: Monotone approximation with linear differential operators. J. Approx. Theory 44, 391–393 (1985) 6. Badea, I., Badea, C.: On the order of simultaneously approximation of bivariate functions by Bernstein operators. Anal. Numé r. Théor. Approx. 16, 11–17 (1987) 7. Mamatov, T., Samko, S.: Mixed fractional integration operators in mixed weighted Hölder spaces. Fract. Calculus Appl. Anal. 13(3), 245–259 (2010) 8. Shisha, O.: Monotone approximation. Pac. J. Math. 15, 667–671 (1965) 9. Stancu, D.D.: Studii Si Cercetari StiintificeXI(2), 221–233 (1960)

Conclusion

During the last 50 years fractional calculus due to its wide applications to many applied sciences has become a main trend in mathematics. Its predominant kind is the newer one of Caputo type with all of its variations because of their easier applicability. These involve usually singular kernels but recently also non singular kernels. In this short monograph we employ an abstract kernel fractional calculus with applications to Prabhakar and non-singular kernel fractional calculi. Our results are univariate and bivariate. In the univariate case we present abstract fractional monotone approximation by polynomials and splines, and in the bivariate case we give the abstract fractional monotone constrained approximation by bivariate pseudo-polynomials and polynomials. This monograph’s results are expected to find applications in many areas of pure and applied mathematics, especially in fractional approximation theory and fractional differential equations. Other interesting applications can be in applied sciences like geophysics, physics, chemistry, economics and engineering, etc. The advantage of this abstract approach is the wide range of its applications in covering the various natural phenomena.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 G. A. Anastassiou, Abstract Fractional Monotone Approximation, Theory and Applications, Studies in Systems, Decision and Control 411, https://doi.org/10.1007/978-3-030-95943-2

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