Topics in Polynomials of One and Several Variables and Their Applications: Volume Dedicated to the Memory of P. L. Chebyshev (1821-1894)

626 Pages·1993·195.874 MB·English

by  Themistocles M. Rassias, Hari M. Srivastava, A. Yanushauskas

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TOPICS IN
POLYNOMIALS OF ONE
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World Scientific
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Printed in Singapore by Continental Press Pte Ltd
PREFACE
m
o Pafnutii Lvovich Chebyshev was born on May 16, 1821 at Okatovo
c
c.
entifi iPne terthsbe urgK oanlu gDae cermegbieorn 8, o1f8 94t.h e Soviet Union and died in St.
ci
s
d He was nominated in 1853, as a junior academician of the St.
orl
w Petersburg Academy of Sciences and was awarded with the Chair of
w.
w Applied Mathematics. The chairs for pure mathematics at the Academy
w
m were then occupied by P. H. Fuss (1798-1855), a great grandson of Euler,
o
d fr M. V. Ostrogradskii (1801-1862), and V. Ya. Bunyakovskii (1804-1889).
dey. Chebyshev is considered to be the creator of the largest prerevolution-
wnloase onl ary School of Mathematics in Russia. Some of the most important members
ou of that School were A. N. Korkin (1837-1908), A. V. Vassiliev (1853-1929),
ns Donal A. A. Markov (1856-1922), A. M. Lyapunov (1857-1918), D. A. Grave
os
heir Applicati19/16. For per An(11u8.8 4m6N7b3 .ue -r1Kn 9trti3ylh 9leho)oiv,s r yVd( 1e.( 8apA6trh.i3 m-S(1het9e-en4k wu5lom)av. sb Cn(e1reh 8vet6ebhr4ye -mso1hr9aee2rvmr6 i))ew,,d aGa)s,p .e apnFcrotr.i ixvcViehmo inrianogt ni oSmonita . n(t1yhP8 eef6otie8eryrl-ds1, sb9,pu 0rrn8oga)b m, afrabeonlimyld­:
es and T8 on 06/ imtya tthheemorayt,i cdailf faenreanlytisails gaenodm "eptrrya,c tkicianle mmaattihcse,m aast iwcse"l.l aHs em haandy ap rkoebenle msesn soef
abl7.5 for the relationship between pure and applied science. In probability theory
Vari2.20 he insisted on rigorous definitions which led Markov to the Markov chains
eral 9.13 of random variables (1906). These chains have proved to be very important
v0
d Seby 1 in statistical physics, genetics, economics and in many other fields [See, for
n example, D. J. Struik, A Concise History of Mathematics, Fourth Revised
a
ne Edition, Dover Publications, Inc., New York, 1987]. In fact, this work of
O
of Markov has proved instrumental in A. N. Kolmogorov's subsequent rigorous
s
al foundation of modern Probability Theory.
mi
no As evidence of Chebyshev's wide-ranging recognition in the interna­
y
ol tional mathematical community, he was elected member of the following
P
s in Academies: Corresponding Member of the Societe Royale des Sciences de
c
pi Liege and of the Societe Philomathique in 1856, of the Paris Academy of
o
T Sciences in 1860 and a Foreign Member in 1874 (the first Russian since
Peter the Great), as well as a Corresponding or Foreign member of the
Berlin Academy of Sciences (1871), the Bologna Academy (1873), the Royal
Society of London (1877), the Italian Royal Academy (1880), and the
V
vi
Swedish Academy of Sciences (1893).
Chebyshev maintained close contacts with many of the greatest
Western European scientists of the time and especially with C. Hermite
(1822-1901), J. Bertrand (1822-1900), L. Kronecker (1821-1891), E. C.
m
o Catalan (1814-1894), and later, also with F. E. A. Lucas (1842-1891) and
c
ntific. SCy. lAve.s tLear is(a1n8t1 4(1-1884917-1),9 2J.0 )L. ioCuhveibllyes h(e1v8 0h9a-d1 8a8l2so), cEo.r rLesinpdoenldoef n(c1e8 7w0i-th1 9I4, 6J).,
e
sci and P. G. L. Dirichlet (1805-1859) [See, for example, A. P. Youschkevitch,
d
orl P. L. Chebyshev, in: Dictionary of Scientific Biography (Ed. C.C.Gillispie),
w
w. Vol. 3, pp. 222-232, Charles Scribner's Sons, New York, 1971].
w
m w This commemorative volume contains a series of scientific articles dedi­
ded froy. rcealtaetde dt ot o Ptahfen uwtioi rkL voofv iCchh ebCyhsehbeyvs, hdevee. pTenh eoseu r aurtnicdleerss,t awndhiicnhg aorfe soinmdee eodf
ownloause onl othfe o nceu rarenndt sreevseeraarlc hv apriraobblleesm asn da ntdh etihre oarpipelsi ciant imonasn.y topics in polynomials
s Dnal It is our pleasure to express our deepest appreciation to all the
no
os
Applicati6. For per stShcceiiee nnstutiipfsitecsr bwP uhabosl sicissohtnainntrgci ebC uiotne. deh datoist itpnhrgios vavindodel udcm. oem. pFoisniatilolyn, wthea tw itshhe tsot aafcf konfo wWleodrglde
eir 9/1
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d T06/ Th. M. Rassias
es an8 on H. M. Srivastava
abl7.5 July 1991 A. Yanushauskas
Vari2.20
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CONTENTS
m
o Preface v
c
c.
ntifi
e On the Characterization of Chebyshev Systems and on Conditions
ci
ds of Their Extension 1
worl Y. G. Abakumow
w.
ww Characterizations for the Existence of a Solution to the Moment
m Problem on a Finite Number of Intervals 9
o
d fr Wm. R. Allaway & X. Liu
dey.
wnloase onl SPoomlyen oRmeisaullst s on Compositions of Algebra-Valued Abstract 35
ou
s Dnal A. M. At Rashed & N. Zaheer
no
os
Applicati6. For per DiscoAn. tinBuaocuosp oAulltoesr nation from a Singularity 43
eir 9/1 Some Inequalities for Polynomials 57
h1
d T06/ M. Bidkham & K. K. Dewan
es an8 on Rate of Convergence of Linear Mean Subsequencies of Fourier Sums 65
abl7.5 TV. K. Bliev & L. P. Falaleev
al Vari132.20 On Markov and Sobolev Type Inequalities on Compact Sets in E" 81
er9. L. P. Bos & P D. Milman
v0
e1
d Sby Application of Chebyshev Polynomials to Antenna Design 101
n
a
e J. L. Brenner
n
O
of A Set of Research Problems in Approximation Theory 109
s
al E. W. Cheney & Y. Xu
mi
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n On Lagrange Polynomial Quasi-Interpolation 125
y
ol
P C. K. Chui, X. C. Shen & L. Zhong
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cs i The Convexity of Chebyshev Sets in Hilbert Space 143
pi
To F. Deulsch

Trigonometric Symmetries: Four-Dimensional Identities of
Modified Chebyshev Polynomials 151
M. Dombroski
vii
viii
On the Completeness of Orthogonal Polynomials in Left-Definite
Sobolev Spaces 173
W. N. Everitt, L. L. Liltlejohn & R. Wellman
Inequalities for Polynomials and Trigonometric Polynomials
m
co Related to the Bernstein Inequality 197
c.
ntifi T. G. Genchev
e
sci Miscellaneous Problems Solved in Terms of Chebyshev's
d
orl Orthogonal Polynomials of the First and the Second Kind 209
w
w. C. C. Grosjean
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w
m A New Method for Generating Infinite Sets of Related Sequences of
o
d fr Orthogonal Polynomials, Starting from First-Order Initial-Value
dey. Problems 247
wnloase onl C. C. Grosjean
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s Dnal Some Remarks for the Methods to Find All the Zeros of a
no
os Polynomial Simultaneously 273
heir Applicati19/16. For per InterMNpo.. l aJIthgouarnryaj hsPuhnri owpaelrat,i esJ . oPf rCahsaedb y&sh eAv. PKo. lyVnaormmiaa ls 287
d T06/
ables an7.58 on aOnrdth Hogeounna l Polynomials on n-Spheres: Gegenbauer, Jacobi 299
Vari2.20 E. G. Kalnins & W. Miller, Jr
eral 9.13 Rational Approximations: A Tau Method Approach 323
v0
d Seby 1 H. G. Khajah & E. L. Ortiz
n
e a On the Rational Chebyshev Approximants to a Real-Valued
n
O Function with an Unbounded Number of the Poles 335
of
s R. K. Kovacheva
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mi
o Orthogonal Polynomials and Ordinary Differential Equations 347
n
y
ol A. M. Krall
P
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s i Theory and Applications of Dickson Polynomials 371
c
opi R. Lidl
T

Some Bilinear Formulas and Integral Equations for Chebyshev
Polynomials 397
R. F. Millar
ix
On some Turan's Extremal Problems for Algebraic Polynomials 403
G. V. Milovanovic, D. S. Mitrinovic & Th. M. Rassias
Extremal Problems for Polynomials and Their Coefficients 435
m G. V. Milovanovic, I. Z. Milovanovic & L. Z. Marinkovic
o
c
c. An Application of the Chebyshev Integral Inequality 457
ntifi D. S. Mitrinovic & J. E. Pecaric
e
ci
ds Some Recent Advances in the Theory of the Zeros and Critical
orl
w Points of a Polynomial 463
w.
w Th. M. Rassias & H. M. Srivastava
w
m Artificial Intelligence Today 483
o
d fr G. C. Rota
dey.
wnloase onl The WR. orRk ooyf Chebyshev on Orthogonal Polynomials 495
ou
s Dnal Matching Polynomials and Holographic Neural Networks 513
no
os
eir Applicati9/16. For per APo ClyenWrot.am inia SlcFsh aemmiplyp of Generating Functions for Classical Orthogonal 535
d Th06/1 H. M. Srivastava
es an8 on A Class of Weight Functions that Admit Chebyshev Quadrature 563
abl7.5 J. L. Ullman
Vari2.20 On Some Applications of Polynomials in the Theory of Integral
eral 9.13 Transforms 573
v0
e1 N. Virchenko
d Sby
an Mean Number of Real Zeros of a Random Trigonometric
e
On Polynomial. II 581
of J. E. Wilkins, Jr
s
al
mi Orthogonal Polynomials of Many Variables and Degenerated
o
yn Elliptic Equations 595
ol
P A. Yanushauskas
n
cs i Linear Stationary Second-Degree Methods for the Solution of
pi
To Large Linear Systems 609

D. M. Young & D. R. Kincaid
A Theorem on Algebra-Valued Pseudo Polar-Derivatives 631
N. Zaheer
TOPICS IN POLYNOMIALS OF ONE AND SEVERAL
VARIABLES AND THEIR APPLICATIONS (pp. 1-7)
edited by Th. M. Rassias, H. M. Srivastava and A. Yanushauskas
© 1993 World Scientific Publ. Co.
m
o
c
c.
ntifi
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ci
s
d
worl ON THE CHARACTERIZATION OF CHEBYSHEV
w.y. SYSTEMS AND ON CONDITIONS OF THEIR EXTENSION
wnl
wo
m se
ou
d frnal
eo
ds
wnloaor per Y. G. Abakumow
s Do16. F
cation06/19/ the cWlosee dc oinnsteidrvera l th[ae, sbp]a -c et hoef scpoanceti nCu[oau,s b ]f u-n cwtiiothn st,h ew mhiecthr ica rpe, ddeeffiinneedd wonit h
AppliE on the help of the equality
s and Their OF SCIENC and with the help of tph(eu in,uo2r)m= ||&t€m||[ aa=,xb ] p \(u0i,( /t)) . - u2(t)\
bleE
aT Later by lin F we shall denote the linear span of the set F C C[a,6].
eral VariNSTITU TChhee bfyisnhiteev sseyt soterm t,h eif saynsyte mel em{ue,n(tt )}f("t)_ 0, €w hhenr{eu ;u};("t_f)0 ) Gh aCs [an,o6 ]m, oisr ec athllaend na
SevN I zeros in [a,b] (if f(t) £ 0).
d N
nA Below we shall formulate the hypothesis about the characterization of
ne aZM Cheyshev systems and shall concern problems, connected with this hypoth­
OEI
of W esis and with the methods, which appear at its basis.
als by This hypothesis is connected with the problems on the extension of
mi
o Chebyshev systems and so at first we shall explain what we understand
n
y
ol about this.
P
s in The Chebyshev system {UJ(0}?=O on ia> &\ ^S called non-extending over
pic the bounds of [a, b], if for any closed interval [c,d\, [a, b],[c,d\ ^ [a, 6] any
o
T system {vi(t)}" , v,(<) £ C[c,d\ such that «,-(<) = i>,(<), for t e [a, b], is
=0
Chebyshev system on [c,d\.
Otherwise the system {u,(2)}"_ is called extending over the bounds
0
of [a, b].
l