223 45 3MB
English Pages [80] Year 1968
Table of contents :
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 ..... 43
9. On the Zeros of Biorthogonal Polynomials...... 49
10. On the Derivatives of Biorthogonal Polynomials 58
11. Conclusion. .......... 65
Bibliography ........................ 67
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 MathematicianProgrammer 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 fulltime 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 nonvanishing of the determinant i, j = 0, 1, 2,..., n1, 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®
242260).
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), 14).
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^(lx) ^ ,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,..., n1 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) 10
(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), 423428). For kz2, the above relations demand that c?Ci o(, q(x) (xa) . (x+a) ^ (xb) , (x+b) and p(x)= x(xa)
, (x+a)
, (xb)
• (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,,., n1
L o
i=n
and (2)
J^p(x)U (x)x^^dx  ('0
for i  0, 1, 2,.., n1 i= n
where p(x) is a nonnegative 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 Xrays 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 selforthogonal 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, pseudoJacobi, 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 k41 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 k1 / . / ones, i.e. f f where the scalars c a . not all zero, i1 ^ ^ 1 then k1 (^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
i1*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
k1 (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 iS
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,..., k1
i= k
and (6)
for 1 = 1, 2,..., k1 ^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,..., k1
(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,.,, k2*
Consider 0  (^k>^kl ^ " ^^k>
'^kl, j S j ) J—

k  1
y
'k1,j (yk'Sj)
j=l = Â.1, S
i
n
c
(fk'Sk)fO*
e
i
t
k1 (4k'Skl)
follows
(^k'8kl)= 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 Ji
/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
^^k1^^
*....... (Sk1* fk^ k
(fl^gl)...........(f,6k) (9)
J 3
=det
(^k1'§][)•••••••• (fk^l* 8^ ) ^ Clearly (^k,gi)^0, For i = k we have
j (fi,^k)=^0 for i = 1, 2,.., k1.
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
j1 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,.., n1
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,.., n1 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,nl detGj^ = det
\
f fl,nl
1, o
^n1,o
^nl,niy
is nonvanishing 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,nl 1
^1,0 ..... . ...... ••• In,l
^l,nl"•*• r(x)....
13
(1 1 )
S ^ ( x ) =.
■^0,0
10.1
^ o ,n
Il,o
11.1
ll,n
fn1,0
fn1,1
1
• ^nl,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
i1, 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 \ ml ' ^
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 llf2
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
/IfS^(f) ll, taken over all
Variation
^
jSS I 12 ^ j  n1 ).
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 .
XY 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^BAI 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
ko.
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 // ^ / ^
/ i1
Le t

/ i1
/
n=i
iJ.
Y n and some
N>0
(/o
Consider n=i
ince and
IX ,1y 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=
^ k0
'' i=mHl
/
k=mfl
23
Clearly,
Â ®
if ^ a
^ I? ■^ ~ (a. I must tend i=itnl ^
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.
UfSn 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 nonnegative 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 c1 )  « (/< ha ) (/6chb )
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°^(lx)^
x^dx =/ 0
°
^0
1=0,
1, 2,.., n1
i=n
When