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Proceedings of the International Conference on Mathematical
Analysis and its Applications, Kuwait, 1985
Edited by
S. M. MAZHAR,
A. HAMOUI and N. S. FAOUR
Kuwait University, P.O. Box 5969
Safat, Kuwait
PERGAMON PRESS
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First edition 1988
Library of Congress Cataloging in Publication Data
International Conference on Mathematical Analysis and
its Applications (1985 : Kuwait, Kuwait)
Mathematical analysis and its applications.
(KFAS proceedings series ; v. 3)
1. Mathematical analysis—Congresses. I. Mazhar,
S. M. II. Hamoui, A. (Adnan) III. Faour, N. S.
(Nazih S.) IV. Title. V. Series.
QA299.6.1565 1985 515 88-4055
British Library Cataloguing in Publication Data
International Conference on Mathematical
Analysis and its Applications (1985 :
Kuwait)
Mathematical analysis and its applications.
1. Calculus
I. Title II. Mazhar, S. M. III. Hamoui, A.
IV. Faour, N. S. V. Series
515
ISBN 0-08-031636-0
In order to make this volume available as economically and as
rapidly as possible the author's typescript has been reproduced
in its original form. This method unfortunately has its typo
graphical limitations but it is hoped that they in no way distract
the reader.
Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
ORGANIZING COMMITTEE
Professor Fuad S Mulla
Professor S M Mazhar
Dr Adnan Hamoui*
Dr Waleed Deeb
Dr Nazih S Faour
Dr R Younis
Dr M N Al-Tarazi
Dr M Al-Zanaidi
* Also representative of the Kuwait Foundation for the Advancement of
Sciences
EDITORIAL COMMITTEE
Professor S M Mazhar
Dr Adnan Hamoui, Dr Nazih S Faour
ORGANIZING COMMITTEE
Professor Fuad S Mulla
Professor S M Mazhar
Dr Adnan Hamoui*
Dr Waleed Deeb
Dr Nazih S Faour
Dr R Younis
Dr M N Al-Tarazi
Dr M Al-Zanaidi
* Also representative of the Kuwait Foundation for the Advancement of
Sciences
EDITORIAL COMMITTEE
Professor S M Mazhar
Dr Adnan Hamoui, Dr Nazih S Faour
PREFACE
An international conference on "Mathematical Analysis and its
Applications", sponsored jointly by Kuwait University and the Kuwait
Foundation for the Advancement of Sciences, was held in Kuwait from
February 18 to February 21, 1985. It was attended by a large number of
mathematicians from all over the world. Twenty one invited talks were
delivered by eminent mathematicians and about 53 research papers were
presented in the form of short communications.
The present volume contains the texts of some of the invited talks and
research papers presented at the Conference. All articles appearing in
these proceedings were duly refereed with the exception of invited talks.
It is a pleasure to express, on behalf of the Department of Mathematics,
sincere thanks to Kuwait university and the Kuwait Foundation for the
Advancement of Sciences for providing necessary funds towards holding the
conference and the publication of its Proceedings.
We acknowledge our indebtedness to Dr Bader Al-Saqabi, ex-Chairman of
the Department of Mathematics, for placing resources of the Department at
our disposal. We also express our gratitude to the invited speakers for
giving stimulating talks at the conference, and to all participants for
making the conference a success.
Finally the staff of Pergamon Press merits our warm thanks for
publishing the proceedings.
S M Mazhar
Adnan Hamoui
Nazih S Faour
1 May 1987
vii
AN ESTIMATE FOR THE RATE OF
CONVERGENCE OF A GENERAL CLASS OF
ORTHOGONAL POLYNOMIAL EXPANSIONS
OF FUNCTIONS OF BOUNDED VARIATIONt
R. BOJANIC*
Department of Mathematics, Ohio State University,
Columbus, OH 43210, USA
1. Let {p (x)} be a sequence of orthogonal polynomials generated by a
non-negative weight function ω(χ) on [-1 , 1 ] and
1
h = [ ω(*)ρ*(£) dt. (1.1)
n
-1
We denote by
%(f) = (]/hn) \ "it) f(t) p (t) at ,
n
-1
tt= 0,1,2,... , the Fourier coefficients of / with respect to {p (x)} and
by
lak(f)pk(x) (k> 0) ,
the Fourier expansion of /. The n-th partial sum of this series will be
denoted by
η
S (f,x) = J a (f)p (x).
n k k
k=o
In 1952, G. Freud proved in [11] that under certain hypotheses about
the weight function ω(χ) it is possible to conclude that
lim Sn(f9x) = f{x) (1.2)
ft-*oo
Invited talk.
The author is grateful to the Kuwait University and the Kuwait Foundation
for the Advancement of Sciences for the support of his participation at
the Conference on Mathematical Analysis and its Applications.
1
2 R. Bojanic
if / is a function of bounded variation on [-1,1], continuous at the point
#€(-1,1). Freud1 s result in its simplest form can be stated as follows.
Suppose that the weight function ω(α?) satisfies the following condi
tions on (-1,1):
(i) ω(χ) < M{\-x2)~A ,
(ii) \p {x)\ <M{\-x2)~BhU2 .
Suppose further that / is a function of bounded variation on [-1,1].
Then (1.2) holds at every point x€ (-1,1) where / is continuous.
This result is clearly the analog of the well-known Dirichlet-Jordan
test for the convergence of ordinary Fourier series of 2π-periodic functions
of bounded variation.
We shall consider here the problem of estimating the rate of conver
gence of the sequence of partial sums {s (f *x)} when / is a function of
bounded variation on [-1,1], not necessarily continuous at the point
x € (-1,1).
In order to obtain the rate of convergence we have to assume that, in
addition to (i) and (ii) , the sequence of polynomials {p (x)} satisfies the
condition
i f i chU1
(iii) J u{t) {t)dt\ <-£- .
Vn
-1
Our main result can be stated as follows.
THEOREM 1. Let {p (#)} be a sequence of orthogonal polynomials on [-1,1]
generated by a non-negative weight function (ύ(χ) and let h be defined by
(1.1). We assume that for every χ € (-1,1) and n — 1,2,... conditions
(i), (ii) and (iii) are satisfied.
If / is a function of bounded variation on [-1,1] then
n
K(f,*)-t(/U 0) (,-0))|<££> 1 V^Y^'ligJ
+ +/
+ \ |/(* + 0) -f(x-O) | \S{$,x)\. (1.3)
n x
Here, g is defined by
'fit)-f(x-o) if -i<t<x<i
gW = { 0 if t = x (1.4)
x
f{t)-f(x + 0) if -\<x<t<\
and ψ is the special function ».
X -1 if t<x
ty{t) = sign(t-x) = \ -0 if t=x
x (K5)
1 if t > x .
Orthogonal Polynomial Expansions of Bounded Variation 3
Also, C(x) > 0 for x £ (-1,1) and V (f) is the total variation of the func
tion / on [a,b ] .
If f is, in addition, continuous at the point x € (-1,1), inequality
(1.3) becomes
\S (f,x)-f(x) <— I V . (f) ·
n M+ w
n n u — ( i +) j\
k=\ x x
Theorem 1 shows that the convergence of the orthogonal expansion of a
function of bounded variation, at a point of discontinuity, depends essen
tially on the behaviour of the orthogonal expansion of the special function
ψ (t) = sign(t-x) at the point t=x .
It is easy to see that condition (iii) is satisfied if and only if the
Fourier coefficients of ψ with respect to {pn(x)} satisfy the inequality
n h
n
It is interesting to observe here that this condition, or condition
(iii), which we had to add to Freud's conditions (i) and (ii) in order to
obtain an estimate of the rate of convergence, is itself related to the
2
rate of best L -approximation of the function ψ by polynomials. Using
orthogonality property of the sequence {p (x)} we have, for any polynomial
Q of degree <n- 1 ,
n 1 W * « ' s ίχ ω(*) **(*>?«(*><**
-1
f ω(*)(φ (ί)-β _ (*))ρ2(ί)4ί .
Λ κ ι
-1
Hence
\l/2 / 1 vl/2
Καη{^χ) < (J ω(*) (**(*)-Vi(t)) dt) (l ^Pn{t)dt)
-1 -1
Taking infimum over all polynomials of degree <n- \ we find that
Κ\αη(*χ)\ <Α^2£^2)(ω.« ),
Γ
(2)
where En (ω»Ψχ) is the constant of best L2-approximation of ψ by poly
nomials of degree <n-l. Thus, condition (iii) will be satisfied if
£un( 2)(^ω>ψ,χψ' ) ^< -η .
Theorem 1 is a very useful tool for the study of convergence properties
of specific orthogonal expansions of functions of bounded variation.
4 R. Bojanic
A particular and yet sufficiently general system of orthogonal polyno
mials to which Theorem 1 is applicable is the system of Jacobi polynomials
l{Pn ' (x) } generated by the weight function
ω(χ) = (1 -xf (1 +x)3
(a ß) fn + a\
and normalized so that P * ( 1 ) = I I . We have here
n \ n /
(α,β) 2α+3+1 Γ(η+α+1) Γ(η+β+1)
Α
η 2η+α+3+1 Γ(η+1) Γ(η+α+$+1)
and, as it is easy to see,
. , (a, 3) a+3 ,, ,v
Ί
lim nh = 2 . (1.6)
n-»<*>
In case of Jacobi polynomials, properties (i), (ii) and (iii) can be
easily established. An estimate for the rate of convergence of the sequence
{Sn(4> >%)} to zero, following a suggestion by R. Askey, can be obtained as
follows. Since x is an interior point of [-1,1], we can first use the
equiconvergence theorem for Jacobi series (see [12], Th.9.12, p.244) to
conclude that
lim Sn^x>x) = 5 Vx (χ+ °) + Ψ (*-°))= ° >
n ->°° χ
which is equivalent to the statement that
oo
It follows then that
oo
Sn^x>*)=- l αα,, ί(ψψ )) PP,^('pa' 'e) (x)
and an estimate for S (ψ , χ) can be obtained by using asymptotic properties
of Jacobi polynomials.
We have the following estimate of the rate of convergence of the
Fourier-Jacobi expansion of a function of bounded variation.
THEOREM 2. Let Ϊ? ' (#)f be the sequence of Jacobi polynomials and let
/ be a function of bounded variation on [-1,1]. I_f_ a >- ^ anc* 3 > ~ 2»
we have
I C(a,3;a0 V x + (\-x)Ik ,
?„(a,3)(f,Ä) -|(f(x + 0) +f(x-0)) v
^ ) γ (q)
k=i
M(a,$;x) ,
+ ]f(x + 0) -f(x-O)
