An investigation of biorthogonal polynomials

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AN INVESTIGATION. OF BIORTHOGONAL POLYNOMIALS

Dissertation

Submitted in Partial Fulfillment

of the requirements for the

degree of

DOCTOR OF PHILOSOPHY (Mathematics)

at the

POLYTECHNIC INSTITUTE OF BROOKLYN

by

Winchung Alvin Chai June 1968

Approved : 7^ Ay

a /

1968

Copy No. fiead of?7Department

P ro Q u e st N um ber: 27733108

All rights reserved INFO RM ATION TO ALL USERS The quality of this reproduction is dependent on the quality of the copy submitted. in the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion.

uest P roQ uest 27733108 Published by ProQuest LLC (2019). Copyright of the Dissertation is held by the Author. Ail Rights Reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346

il Approved by the Guidance Gommitte

Major :

Numerical Analysis

'S tanie y Fre i ser Associate Professor of Mathematics

Minor :

Functional Analysis

Harry Hpehstadt Professor of Mathematics Head of Department

Minor ;

Differential Equations

Burton B, Lieberman Assistant Professor of Mathematics

ill

Microfilm or other copies of this dissertation are obtainable from the firm of University of Microfilms 313 N. First Street Ann Arbor, Michigan

iv

VITA V/inchung Alvin Chai was born in Hunan, China, on August 21, 1939.

He came to U. S. with his parents in the

winter of 1955, entered Wittenberg College, Springfield, Ohio in 1956, where he received A. B. degree in 1960. While working as Mathematician-Programmer Analyst at General Precision's Aerospace Research Center, Mr. Chai attended the New York University Graduate School of Arts and Science and received M. S. degree in Mathematics in 1964.

He also taught part time at Stevens Institute of

Technology.

Prior to joining General Precision, Mr. Chai

was an analyst in Operation Research at Bell System. Mr. Chai passed his preliminary oral examination in September, 1967 and worked full-time on this thesis from September, 1967 to May, 1968.

Dedicated to My Parents

VI

ACKNOWLEDGEMENTS I would like to thank my adviser.

Professor Stanley

Preiser for the invaluable advise and guidance he gave so freely to me while I was a student as well as a Doctorate Candidate.

I would also like to thank Professor Hochstadt

for his guidance on my work and Professors Chester and V

Lieberman

for their advises.

Finally, I would like to take this opportunity to thank Mr. Henry Chai, who has always been more than just a brother.

Without his encouragement and help over the

years, it would be impossible for me to complete my graduate studies.

v ii

AN ABSTRACT AN INVESTIGATION OF BIORTHOGONAL POLYNOMIALS by Winchung Alvin Chai Adviser: Stanley Preiser Submitted in partial.fulfillment of the requirements of the degree of Doctor of Philosophy (Mathematics)

Let X be the Hilbert space L ^ ja, l^^n^ ,

nj

for n ^ m ,

and

.

Two sequences

X will be called biorthogonal if (^^y 0 for n==m.

The purpose of this investigation

was to determine properties of biorthogonal sequences and to investigate those properties that are analogues of the properties of orthonormal set. Given

nj , a necessary and sufficient

condition for the existence of unique biorthogonal sets is the non-vanishing of the determinant i, j = 0, 1, 2,..., n-1, for n = l ,

2,...

9j)|, When

^

are polynomials, this result reduces to a theorem on the uniqueness and existence of biorthogonal polynomials which was previously proved by Konhauser. 11(1965),

(J. Math. Anal. Appl®

242-260).

The biorthogonal sequence

said to be

v iii

complete if each of the system X.

It is normalized if

each of ^//^n.//|

complete in = 1.

It is bounded if

Is bounded.

For a complete, bounded

and normalized biorthogonal sequence, 3

constants M, M q > 0

such that GXD

Mo(f,f)6: 5 1

/(f,/^)|^é:M(r,f) V'feX.

In particular, if

nr:l X r:

^

lljthen the least upper bound of

II

f-

(f

It ,

taken over all f(x) such that the total variation of f(x) is 1, tends to zero not faster than n~ sequence, this result

reduces to theorem which was

established by W. Rudin (Duke Math. Let (f,g)zr f

For the orthonormal

J» 19(1952), 1-4).

p(x) f (x) g (x)dx where p(x) is a non-

negative admissible weight function,

is some closed

interval in the real line or a closed curve in the complex p lane. V/hen f^cx^, g ^=x^^, p(x)= x^(l-x) ^ ,o(> -1,

non­

negative integers, and /% (a, b), the biorthogonalization process determines a pair of biorthogonal polynomials I^Z^(x^)^ , ^T[j^(x)^ in x^ and x respectively.

These

polynomials are generalizations of the Jacobi polynomials and possess many properties which are analogues of the properties of the Jacobi polynomials. in the complex plane, p(x)= x^“

For /%unit circle where a is a positive

integer and b :)^0, it leads to a pair of biorthogonal polynomials which are generalizations of the Bessel polynomials.

The polynomials in each of the above sets

IX

satisfy a pure recurrence relations connecting k f 2 successive polynomials.

For the polynomials in x^, they

satisfy a differential equation of order k + 1

of the form

where both the differential operator scalardepend

on n.

However,

equation reduces

to the equation

and

for k - 1, the differential for the Jacobi and Bessel

polynomial respectively© If r is restricted to a closed interval (a, b) in the real line, and if

are a pair of

biorthogonal polynomials in

|a, b^ , then the n zeros of

H^(x) are all distinct and interior to (a, b) and the zeros of Xln+i^^) and H^(x) are mutually separated.

In the case

of Zn(x^), it possesses exactly n simple roots in (a, b) if 0 is not a root interior to (a, b). n distinct roots

in (a, b) with 0

multiplicity k.

All

Otherwise Zj^(x^) has

as a repeated root of

these zerosare mutually separated for

any two consecutive Z^(x^)'s. If the derivative ll^(x) of H^(x) also satisfi* biorthogonal condition

J

i = 0, .1, 2,..., n-1 0

i 2 n,

then the admissible weight function q(x) must satisfy the differential equation ( 2 Ü a^_x^^)q» (x) = K(x^^-j2/^)x^”^q(x) 1-0

(K,

■a. ,are constants) 1

with the boundary condition q(a)% 0. It is to be noted that the solution q(x) also satisfies

the boundary condition q ( b ) - 0 .

The admissible weight

function p(x), now can be determined, up to a multiplicative constant, from q(x) by the relation p(x}=.

For kzl,

^

q(x)

.

the above result reduces to the orthogonal

case which leads to the Jacobi polynomials.

This result

was first established by.W© Hahn and later investigated by H. Krall (Bull. Amer. Math. Soc. 42(1936), 423-428). For kz2, the above relations demand that c?Ci o(, q(x)- (x-a) . (x+a) ^ (x-b) , (x+b) and p(x)= x(x-a)

, (x+a)

, (x-b)

• (x+b) are constants > 0).

XI.

table: of

contents

■ PAGE 1*

Introduction..... .............. ................

1

2.

General Theorey....... ........... .......... ..

5

3. On the Mean Convergence of Biorthogonal Expansi o

h

s

15

4. An Extension of the Generalized Hypergeometric Functions........................................

25

5. Biorthogonal Polynomials Suggested by Jacobi Po lynomi als.....................................

27

6. Biorthogonal Polynomials Suggested by Laguerre Polynomials • ...................................

35

7. Biorthogonal Polynomials Suggested by Hermite Polynomials.....................................

39

8. Biorthogonal Polynomials Suggested by Bessel Polynomials ..... 9.

43

On the Zeros ofBiorthogonal Polynomials......

49

10.

On the Derivatives of Biorthogonal Polynomials

58

11.

Conclusion.

..........

65

Bibliography........................

67

1. Introduction. The general objective of this thesis is to investigate biorthogonal sequences in a Hilbert space, in particular, to extend the notion of classical orthogonal polynomials of Jacobi, Laguerre, Hermite and Bessel to two sets of polynomials in (1)

/p(x)Z^(x^)x^dx = r0 '

^ ;

and x respectively such that for i -0, 1, 2,,., n-1

L o

i=n

and (2)

J^p(x)U (x)x^^dx - ('0

for i - 0, 1, 2,.., n-1 i= n

where p(x) is a non-negative admissible weight function, I ' is some closed interval in the real line or a closed curve in the complex plane. For k % 1, 2^(x^)= l[j^(x), n=: 0, 1, 2,..., and the above conditions reduce to the orthogonal condition. Although, the notion , of biorthogonal polynomials is known to be discussed as early as 1886 [l), the literature concerned with its properties is quite scarce, [ 4 ],

and consists principally of

andjd].

In [2], A. Erdelyi pointed out that from every Imown orthogonal set of functions or from a biorthogonal set of functions,

a new biorthogonal set»can be formed by

fractional integration by parts.

As an example he

discussed in detail the construction of a new biorthogonal system from the Laguerre functions!

But he did not

investigate general properties of these biorthogonal sets. In [3], L. Spencer and F. Fano, in carrying out calculations involving the penetration of gamma rays through matter,

introduced a pair of biorthogonal polynomial

sets in x and x^.

For the polynomial in x^, they gave

formulas, derived mixed recurrence relations,

and presented

a third order differential equation of the form (3)

LY„r A( x )y ! ' + B{x)Yn"+ C( x )Yjj' = >

where A(x), B(x), C(x) are functions of x independent of n and /\^ is a parameter independent of x. Curiously, the subject of biorthogonal polynomials was not the primary objective of Spencer and Fano's work. They were mainly interested in the expansion of the space distribution of X-rays in a suitable polynomial system. After some extensive numerical computations, they showed ’’good convergence” properties of their methods.

However,

they did not really touch the question of convergence from the analytic point of view, nor did they discuss the property of completeness of these polynomials and the merit of deliberately using non self-orthogonal polynomials for the expansion of arbitrary functions. In (4^, S. Preiser showed that, apart from real linear transformation, there is only one third order differential equation of the above type, from which biorthogonal polynomials are derivable.

For each set of the biorthogonal

polynomials, he established pure recurrence relations connecting four successive polynomials, as well as several

mixed recurrence relations.

He also proved that no first

order differential equation,

and only five second order

differential equation of the above type can generate biorthogonal polynomials.

The polynomials thus derived

are the Hermite, Laguerre,

Jacobi, pseudo-Jacobi, and the

Bessel polynomials. Recently,

J. D. E. Konhauser

[sj considered biorthogonal

polynomials in real polynomials r(x) and s(x).

He

systematically discussed properties which are analogues of properties of orthogonal polynomials,

Included were

necessary and sufficient conditions for the existence and uniqueness of biorthogonal polynomials, a sufficient condition which ensures the existence of pure recurrence relations, and information on the location and number of the real zeros of polynomials.

Konhauser fôj also discussed

the particular biorthogonal polynomials determined by r(x)=x,

s(x) = x^, p(x):^x°e”^, c > -1,

o,cc,).

He showed

that these polynomials possess many properties which are analogues or extensions of the properties of the Laguerre polynomials.

For both sets of polynomials, he established

mixed recurrence relations from which a differential equation of order k4-1 is derived.

For k;= 1, the differential

equation reduces to the equation for the Laguerre polynomial. For k=2, the differential equations reduce to (3) which was discussed by S. Preiser. In this thesis, in addition to investigating the biorthogonal polynomials suggested by the Jacobi and Bessel

polynomials, we shall be concerned with the mean convergence properties of the biorthogonal expansions, the separation properties of the zeros of these polynomials, and the derivative of biorthogonal polynomials.

2.- General Theory. In this section we

develop the general theory

which is both useful and essential in our discussion of biorthogonal polynomials. Vife work in the Hilbert space

[k, 10

.

This is

the space of functions f(t) for which xb J I f(t)| a

(t) < 0> a

exists for i = 0, 1, 2,... Definition 1 .

Two sequences

^g%^ of

ja,

0

will

be called biorthogonal if (f%, gjn) ^ 0

if n f m 0 if n r m

It is clear from the definition that sequences of this kind are necessarily linearly independent©

For otherwise

if 3 ^ir such that f is linearly dependent on t he proceeding k-1 / . / ones, i.e. f f where the scalars c a . not all zero, i-1 ^ ^ 1 then k-1 (^k* ®k^^ i (^i; - 0 1^1

which is contradiction. Definition 2 . and

A pair of finite sequences ^fi,..., f ^

...,

is said to be biorthogonalizable or

admits biorthogonalization if there exists a pair of biorthogonal sequences

such that

[ ^i=l (1)

L J i%l

i-1*1

1—1

where 7v

are scalars with the property that 4^ 0 f

^k ^

for k — 1, 2,,,,, n#

The following Lemma is a direct consequence of the definition, therefore we need not give the proof. Lemma.

For any n, f]l^,.,., f

where

Hf

r{ .^1 , ..,

n

r", and

n

..., f^Dl stands for the subspace generated by

Thus for the biorthogonalizable sequence

(1), the

following mixed recurrence representations hold

s k —1

(2)

k+'

A,i

h

k-1 (3)

d, , k,i ' 1

*^k=/^,kS:k4k

(4)

^k.

& k

(5)

^k.

for k — 1, 2, #. #, n. Where

-f

'

^k,iy snd jb^^^Yare scalars

with the property that Definition 5 .

4 Of

for k - 1 ,

A pair of infinite sequences

2,.., n. , -j^SnJ i-S

said to be biorthogonalizable if for any finite n, ^ f ^ , f ^ and

gnj admits biorthogonalization. It is highly desirable to have equivalent condition

for biorthogonality.

In the following theorem we establish

such condition. THEOREM 1 .

A necessary and sufficient condition for a pair of sequences |^n0 »

given in (1) to

form a biorthogonal sequence is that for any k, the (5)

following conditions hold.

°

for

=^0

1=1,

2,..., k-1

i= k

and (6)

for 1 = 1, 2,..., k-1 ^0

Proof :

i :: k

Necessity.

We give the proof of (5).

The proof of (6) is

similar.

We show this by induction on n.

For n - 2 ,

by the biorthogonality of

must have 0 - ’

^

(^2"/^l'1^1^ ' /^1,1

s i n c e i t The fact

follows

(^u, g^)

(fg' 0.

g g )ÿ 0 is certaily clear.

By induction hypothesis, we now assume that (5) is true for all n < k.

Our job is to show

8

(V)

= °

1=1,

to

i ^k

We prove

2,..., k-1

(7) by induction on i.

For i - 1, O = ( 0 k , ^ i ) it follows

S i n c e 0,

(j!)k,gQ_) - 0.

We now assume that (7) is true for

1, 2,.,, k-2*

Consider 0 - (^k>^k-l ^ " ^^k>

'^k-l, j S j ) J—

-

k - 1

-y-

'k-1,j (yk'Sj)

j=l = Â.1, S

i

n

c

(fk'Sk)fO*

e

i

t

k-1 (4k'Sk-l)

follows

(^k'8k-l)= 0

This completes our proof for the

necessary condition. Sufficiency. Consider

i>j

=

°

J7i

In the case i = j, recall from (3) ^

'/'i=A,i si-^ /' ( { ^ i ' ^ i ) = A , i

^i,j

(fi'Si)+ 3 2 : &i,j J-i

-/4,i (fi'Si)fo Therefore

(1) is satisfied, i.e. ) ^ n \ r n [ form a

biorthogonal sequence. In the case of an orthogonal sequence, given an 1 arbitrary sequence j f , 3 an orthonormal set | such that

is some linear combination of |fi^ provided |^fi^

is linearly independent.

For the biorthogonal sequences,

we have the following existence theorem which is similar to the theorem for the classical orthogonal case. THEOREM 2 .

A necessary and sufficient condition that ^ f ^ g 3 j

admits biorthogonalization is that

f or any k . ( (fl; g]_)

(fi; 8%)

det(Jk - det \ (^k* §1^ .............. / is non vanishing. Proof :

Sufficiency.

Define :

/

(Sl'fk) (8)

j)^:=det

^^k-1^^

*....... (Sk-1* fk^ k

(fl^gl)...........(f,6k) (9)

J 3

=det

(^k-1'§][)•••••••• (fk^l* 8^ ) ^ Clearly (^k,gi)^0, For i = k we have

j (fi,^k)=^0 for i = 1, 2,.., k-1.

10

9k = det where

^^(ÿk,gk)+°

is the transpose of Gj^.

/ ! By theorem 1, ^f^^ ,

admits biorthogonalization,

Necessity. We prove this by contradiction. Suppose det G^s.0 for some

0 ^

all zero such that k aj(fi,gi)-0

for i = 1, 2,..., k

.1^1 j' Let ^

be the largest integer such that

and aj2^^ 0 « Define :

W- ^ j'-i

a.g. ^ J

^ -1 , ' g/;=W + 2 T b.g. ^ — 1=1 -i J Clearly (f.,W)- ^

where b , = -a, •) — ^

a.(f^,g.)- % T

j-1 Since

l^k^^ j^Skj

1-

)= 0

' ^i''

j

for i^^ j 0

i- j

consider

S

/?

W

+

^

^

T

for irl, 2,..,k

J

i

h ((j)^,

a.(f.,g )=0

j;l. J

biorthogonalization ^ 3

such that

and

^

—

^

ij

11

since Vil2 1

for 1 = 1, 2,.., n ^ W J.

Y

= 1, 2,.., n

J “i. This is a contradiction.

Hence det

for any k.

Biorthogonal Polynomials. When the sequence ^f^^ and ^ a r e

polynomials.

Theorem 1 and 2 immediately reduce to the theorems which were previously established by Konhauser [b].

However,

before we state them here, we need the following definitions Definition 1 ' .

Let r(x) and s(x) be real polynomials of

degree k > 0 and h > 0 respectively.

Let R^^x) and Sj^(x)

denote polynomials of degree m and n in r(x) and s(x) respectively.

The polynomials r(x) and s(x) are called

basic polynomials. Definition 2 ' . function on

The function p(x) is an admissible weight

if all the moments ^p(x)

exists, with

where

0? (x)j ^ IsCx^'^dx

, i, j = 0, 1, 2,...

b. io,o= J p(x):>o a

is some closed interval in the real line or a

closed contour in the complex plane. THEOREM 1' .

(Konhauser

)

A necessary and sufficient

condition for two sets of polynomials ^R^(x)^ , |sj^(x)^ to be biorthogonal over

with respect

to the admissible weight function p(x) is that

12

J^p(x)R^(x) C s (xQ^dXrj^O

for i=0, 1, 2,.., n-1

r

i;:n

and

r

L

J*p(x) S (x) jj? (x[) ^dx=fG

THEOREM 2'

(Konhauser

)

condition that

for 1=0, 1, 2,.., n-1 irn A necessary and sufficient (x^ ^

, ^|s (x ^ ^

biorthogonalization over

admits

with respect to

the admissible weight function p(x) is that ( fo,o ..... . • •» Io,n-l detGj^ = det

\

f fl,n-l

1, o

^n-1,o

^n-l,n-iy

is non-vanishing for any n. Where I i,j=

^

p(x)

(r[x])^ [s(x)]Jdx,

i, j - 0, 1, 2,...

.The explicit representations of the biorthogonal pair in (8) and

(9) also have a far reaching consequence in the

study of biorthogonal polynomials. |r(x3

g (x) = js(x^

(8) and (9) becqme

^0,0 (10)

R^(x) =

In the case f ^ ( x ) %

-0,1

^o,n-l 1

^1,0 ..... . ...... ••• In,l

^l,n-l"•*• r(x)....

13

(1 1 )

S ^ ( x ) =.

■^0,0

10.1

^ o ,n

Il,o

11.1

ll,n

fn-1,0

fn-1,1

1

• ^n-l,n

s(x)......

. [^s (x]j ^

If we let the variable of integration be Xj_ in the 1th row of (10) and (11) where 1 = 1 ,

2, ••, n,

(10) and (11) I

reduce to

(12)

1

r(x^)........[r(x],j) ^

s(x 2 )

r (x2 )s (xg )•. , [r (xg

(xg ) p(x]_). .p(xjj)dxi-.dx

n

1

(IS)

1

S n (x )

r(xg) a

r(x)

. . .

s(x^) ......................^ (xi3

n

s(xg)r(xg)...... [s (xgTj^r (xg )

»

*

^ |(x^Tj

|r(x^

8(xg) {r(x^^

1

If we factor out

s (x)......

4 0

pCxq). .p(xn)dxi..d

|s(xn)j ^(x^lj • ]^(x^ ^

[s(xj|_0 1“1 from the 1th row of the

determinant in (12) and permute the indices on

i-1, 2,*,,

n in all possible ways in (12), we obtain (after proper interchanging the rows of each new determinants).

14 (14)

1

s ( x i ) , ............

1

8( X2) , ..........

/ ' ■' /

iTT n:

• n

«

1

s ( x ^ ) , ............

1

r ( x ^ ) ..

fU ill»

1

r(x%)..

(rCxg)] p ( x ^ ) . . . ' p (x ^ )d X 3_.

« 1

r(Xj^)..

1

r(x )...

[.(x )]n

These two determinants under the integral sign in (14) are Vandermonde determinants whose factorization is well known.

Applying the factorization, we obtain

(15)

R (x)r '

( ‘ '"f

^

-8(x,|))

i| , jt= l, 2, ..,n

TT

-r(Xj^)) p(x^).. p(xji)dx^. .dXj^

ig j g ““l> 2,.., n*KL In like manner,

(13) becomes

(16)

i^(x)= ' i ’*;/

(r(Xi, ) -r(x. })

.

i.< 0.

^

ijl^,j 1, 2,.., n

TT )p(xi) . op(x^)dXi. .dx^ where

15 3• On the Mean Convergence of Biorthogonal Expansions Preliminary Again, we assume X to be Hilbert space w ith the usual norm, i.e. Definition 1 . in X.

(ifI) ~(f, f

Let

[a,

J jf(^dp a V

^ biorthogonal sequence

The sequence ^

^ ^ ^ J Is said to be normalized

if (^n' ^ n ) " l " Definition 2 .

The biortho gonal sequence

[j^^9

is

said to be complete if each of the systems [*^n^^[^n| is complete in X, that is, the subspace generated by and the subspace generated by

(t/y7

, are each everywhere

dense in X. For a biorthogonal system, the following theorem holds. THEOREM 1 .

(Erdelyi

).

If one of the biorthogonal

systems is complete, then the other also is complete, i.e. r X-

[^n]j

X=

Proof: Suppose X=

•

[Lrnj]

Y^X-

such that Y J ^

Yg X

we can write Y=

tfHen 3

Y + 0,

for all n.

Since

^nrn'

Consider I w, \ 0 . { Y , r j = ( ^ ^njn > L \ m-l ' ^

c/o nz:l

Hence all coefficients a^cO,

Y = 0 which is a

,

16

contradiction.

Thus the theorem is proved.

When the biorthogonal set is a pair of polynomials and

in x and x^ respectively, the above

theorem asserts that the system

is complete if

and only if the system |lU(^(x)| is complete.

In particular,

since the classical orthogonal polynomials

of Jacobi,

Daguerre and Hermite all form a complete set, it follows that the corresponding biorthogonal polynomials in x^, satisfying conditions

(1.1), must also be complete.

(The existence of such biorthogonal polynomials will be discussed later). Definition 5 .

The biorthogonal sequence |^^n^ f

said to be bounded if _j a constant M > 0 and 11

is

such that

^I & M

^ M for all n.

Definition 4 . £a, b] .

Let V(f) be the total variation of f on

Vife say that f e BY [a, ^

if V ( f ) < < ^ '

In this section, we investigate mean convergence of the biorthogonal expansion of the form

s^(f )= ± G.f. 1=1 Our result asserts that if the biorthogonal sequence ^^n^ ,

bounded, complete, and normalized,

then ^

constants M,M q > 0 such that Mq Ilf

^ ^

M llf||2

n =1 As a corollary, we show that a necessary and sufficient condition for a collection of complex numbers ■{

f to

17 determine

j afunction f= %> 0%^^ n=l

(This result isknown and can If X - L ^ j _ o , of

Ï]

, we show

than n

•

of f(x) in [o, ^

X is that

be found in

/If-S^(f) ll, taken over all

Variation

^

jSS I 12 ^ j | n-1 ).

that the least upper bound f(x) such

that the total

is 1, tends to zero not faster

For the orthonormal

sequence, this result was

proved by V\[, Rudin [o]. Proof of Main Results. Let

be a bounded, complete, and

normalized biorthogonal sequence in X. ( (70 Let Y f EX 2 1 j(f, ^ n ) h < < = ° n=l ' Define.

11f 11 y - ^ |(f, n=l

for every f é. Y.

Clearly II Hj is & norm in Y. Lemma 1 .

X-Y and |lf f( ^ 6 M (!f (I ¥ f e X and some

M > 0.

Proof ; First we observe that since independent,

is linearly

an orthogonal sequence

that each

is a linear combination ofthe

n sequence of

the system

.

of X such

Since

first is

complete, it follows that the orthonormal set is also complete. (1) ■

Hence we have for n = l ,

2,..

1— I Solving

(2)

from (1) we obtain a system of the form

forn^l,

2,..

18

Let cK.

0

1,1 cX

2,1

2,2

hi,n

0

/41,1 A,i

A , 2

A,1

/%,2

0 O'

B =. /6 0 / n,n

Clearly (5)

AB^BA-I Let

C = ((

.)) 1,0

S i n c e i s

1, 2,..,

complete, by the Parseval's identity,

we have (A'i^n)--||_ ( A A )

j k=^o for all n.

for k = 1, 2, 3,... x 6 (X, 11

Ily)

such that

V^. I j2 ^ (a^l k=l

I

.

which is uniformly bounded

Hence it also be true as

•

That is

21

I'

m n->^

El I

^

k=o

k=o

ri'^^

I

k-o.

which is the desired result. Next observe that since the system independent,

is linearly

there exists an orthogonal s e q u e n c e o f

X such that each

is a linear combination of the first

n terms of the system \UJ'\ ^ i.e., n I n, i i lor every n. Since

scalars -J

(

is complete and uniformly bounded it

follows that the orthonormal set

j is also complete

and // ^ / ^

/ i-1

Le t

-

/ i-1

/

n=i

i-J.

Y n and some

N>0

(/o

Consider n=i

ince and

IX ,1-y iS'/z. ^ ( ^2 I U -J .2 2 ^ n=l 1=1 o i' ) ^ is uniformly bounded for all n

(Z . i=l

Ia 1

oo , the last equation implies that

//x// < oo

n=l which establishes that x e (X,// //).

Now appealing to

Lemma 1 and the continuity of the inner product, w e can assert that

x

€,(x,ll

lly)

and

22

( Z k=l

& i=l

Finally we show that

(ak . zero since

The second term

llx^^ -xlL^ also tends to

(%nn>^) ~

^

- (x,

as n-^» |a^|^ n=l

n=l

(11 ) If

and cpo J ^2% ^nrn converges to x, then a^ = n=l


_a 6 -Ü /n

^

, it suffices to show that

converges with respect to 11 11 . y

Consider the partial sum

^n'

n h 1=1 and suppose that n > m; then

lls,-sjlf=

^ k-0

'' i=m-Hl

/

k=m-fl

23

Clearly,

 ®

if ^ a

^ I? ■^ ~ (a. I must tend i=-itn-l ^

converges,

^

— 2 to zero and the series ^ fa^l must converge. n=l (ii) This is clear in view of the continuity of the inner product.

THEOREM 4 .

Let

l.u. b.

Uf-Sn o '^r/^n ^ A n ^

)

for

every bounded, complete and normalized biorthogonal set

.

Proof ; Since

is bounded, normalized and complete

it follows from Theorem 2, there exist

constants

K, K > 0 (1)

K

(I f U ^ 4= /1 f/j 2

Let

1%^= l.u. b.

^

Kq

11 f 111

Mf~Snil_v

v f e Lg [o, g .

( 0 < V ( f ) < c ^ ).

V(f ) Clearly

vÇ

^

^ Vk^

\

Let

be an orthonormal set (not necessarily

complete) such that Wj^EBV, W ^ 4 constant. from the definition of

It follows

that

A)' 4 V

24 Now, for the rest of proof, we need only modify Rudin*8 original argument by noting that n

n

n

k=l

i=l

i=l

é.

n M.

where M is a non-negative integer which has the property that II

^A M

V

kz=l, 2, 3.,,

25 4.

An Extension of the Generalized Hypergeometric Functions. Let

, (1)

W=E^ (a,h;'c;g^)-'

where

(&)j(b)kj j=o (c)kj

TT~

- Yr(cii:a -1)) -Z^( A(+ra) TT A ( + r ( b j ^ ) ^ L

where

i=l

/

^

K

/

^

i=lt

W=0

^

az

For r = k = 1,

(3) reduces to

(/< t c-1 ) - « (/< -ha ) (/6c-hb )

W c0

which, by employing the relations y^ = sy^(/{-1 )W-

K

becomes

differential equation.

- zW* and

(2), the ordinary hypergeometric

27

5. Biorthogonal Polynomials Suggested by the Jacobi Polynomials «> The Jacobi polynomials in the interval

may

be written (1)

=

Ey(-n, l4 o ( 4 - + n ;

IM;

x)

satisfying the biorthogonal condition (2) j

x°^(l-x)^

x^dx =/ 0

°

^0

1=0,

1, 2,.., n-1

i=n

When