Topics In Polynomials Of One And Several Variables And Their Applications: Volume Dedicated To The Memory Of P L Chebyshev (1821 - 1894) Volume Dedicated to the Memory of P L Chebyshev (1821 – 1894)

This Volume Presents An Account Of Some Of The Most Important Work That Has Been Done On Various Research Problems In The Theory Of Polynomials Of One And Several Variables And Their Applications. It Is Dedicated To P L Chebyshev, A Leading Russian Mathematician. Contents:on The Characterization Of Chebyshev Systems And On Conditions Of Their Extension (y G Abakumov)on Lagrange Polynomial Quasi-interpolation (c K Chui Et Al.)the Convexity Of Chebyshev Sets In Hilbert Space (f Deutsch)on The Completeness Of Orthogonal Polynomials In Left-definite Sobolev Spaces (w N Everitt Et Al.)a New Method For Generating Infinite Sets Of Related Sequences Of Orthogonal Polynomials, Starting From First-order Initial-value Problems (c C Grosjean)orthogonal Polynomials On N-spheres: Gegenbauer, Jacobi And Heun (e G Kalnins & W Miller, Jr)extremal Problems For Polynomials And Their Coefficients (g V Milovanovi Et Al.)some Recent Advances In The Theory Of The Zeros And Critical Points Of A Polynomial (th M Rassias & H M Srivastava)artificial Intelligence Today (g C Rota)a Certain Family Of Generating Functions For Classical Orthogonal Polynomials (h M Srivastava)mean Number Of Real Zeros Of A Random Trigonometric Polynomial. Ii (j E Wilkins, Jr)orthogonal Polynomials Of Many Variables And Degenerated Elliptic Equations (a Yanushauskas)and Other Papers Readership: Mathematicians And Mathematical Physicists. Keywords:polynomial Inequalities;chebyshev Polynomials;approximation Theory;fourier Series;special Functions;lagrange Polynomials;markov, Sobolev And Bernstein Inequalities;orthogonal Polynomials;generating Functions;holographic Neural Networks;integral Equations;integral Transforms;rational Approximations;elliptic Equations;sobolev Spaces On the characterization of Chebyshev systems and on conditions of their extension / Y.G. Abakumow -- Characterizations for the existence of a solution to the moment problem on a finite number of intervals / Wm. R. Allaway & X. Liu -- Some results on compositions of algebra-valued abstract polynomials / A.M. Al Rashed & N. Zaheer -- Discontinuous alternation from a singularity / A. Bacopoulos -- Some inequalities for polynomials / M. Bidkham & K.K. Dewan -- Rate of convergence of linear mean subsequencies of Fourier sums / N.K. Bliev & L.P. Falaleev -- On Markov and Sobolev type inequalities on compact sets in [symbol] / L.P. Bos & P.D. Milman -- Application of Chebyshev polynomials to antenna design / J.L. Brenner -- A set of research problems in approximation theory / E.W. Cheney & Y. Xu -- On Lagrange polynomial quasi-interpolation / C.K. Chui, X.C. Shen & L. Zhong -- The convexity of Chebyshev sets in Hilbert space / F. Deutsch -- Trigonometric symmetries: four-dimensional identities of modified Chebyshev polynomials / M. Dombroski -- On the completeness of orthogonal polynomials in left-definite Sobolev spaces / W.N. Everitt, L.L. Liltlejohn & R. Wellman -- Inequalities for polynomials and trigonometric polynomials related to the Bernstein inequality / T.G. Genchev -- Miscellaneous problems solved in terms of Chebyshev's orthogonal polynomials of the first and the second kind / C.C. Grosjean -- A new method for generating infinite sets of related sequences of orthogonal polynomials, starting from first-order initial-value problems / C.C. Grosjean -- Some remarks for the methods to find all the zeros of a polynomial simultaneously / M. Igarashi -- Interpolatory properties of Chebyshev polynomials / N. Jhunjhunwala, J. Prasad & A.K. Varma -- Orthogonal polynomials on n-spheres: Gegenbauer, Jacobi and Heun / E.G. Kalnins & W. Miller, Jr -- Rational approximations: a Tau method approach / H.G. Khajah & E.L. Ortiz -- On the rational Chebyshev approximants to a real-valued function with an unbounded number of the poles / R.K. Kovacheva -- Orthogonal polynomials and ordinary differential equations / A.M. Krall -- Theory and applications of Dickson polynomials / R. Lidl -- Some Bilinear formulas and integral equations for Chebyshev polynomials / R.F. Millar -- On some Turan's extremal problems for algebraic polynomials / G.V. Milovanovic, D.S. Mitrinovic & Th. M. Rassias -- Extremal problems for polynomials and their coefficients / G.V. Milovanovic, I.Z. Milovanovic & L.Z. Marinkovic -- An application of the Chebyshev integral inequality / D.S. Mitrinovic & J.E. Pecaric -- Some recent advances in the theory of the zeros and critical points of a polynomial / Th. M. Rassias & H.M. Srivastava -- Artificial intelligence today / G.C. Rota -- The work of Chebyshev on orthogonal polynomials / R. Roy -- Matching polynomials and holographic neural networks / W. Schempp -- A certain family of generating functions for classical orthogonal polynomials / H.M. Srivastava -- A class of weight functions that admit Chebyshev quadrature / J.L. Ullman -- On some applications of polynomials in the theory of integral transforms / N. Virchenko -- Mean number of real zeros of a random trigonometric polynomial. II / J.E. Wilkins, Jr -- Orthogonal polynomials of many variables and degenerated elliptic equations / A. Yanushauskas -- Linear stationary second-degree methods for the solution of large linear systems / D.M. Young & D.R. Kincaid -- A theorem on algebra-valued pseudo polar-derivatives / N. Zaheer On the Characterization of Chebyshev Systems and on Conditions of Their Extension (Y G Abakumov) On Lagrange Polynomial Quasi-Interpolation (C K Chui et al.) The Convexity of Chebyshev Sets in Hilbert Space (F Deutsch) On the Completeness of Orthogonal Polynomials in Left-Definite Sobolev Spaces (W N Everitt et al.) A New Method for Generating Infinite Sets of Related Sequences of Orthogonal Polynomials, Starting from First-Order Initial-Value Problems (C C Grosjean) Orthogonal Polynomials on n-Spheres: Gegenbauer, Jacobi and Heun (E G Kalnins & W Miller, Jr) Extremal Problems for Polynomials and Their Coefficients (G V Milovanovi et al.) Some Recent Advances in the Theory of the Zeros and Critical Points of a Polynomial (Th M Rassias & H M Srivastava) Artificial Intelligence Today (G C Rota) A Certain Family of Generating Functions for Classical Orthogonal Polynomials (H M Srivastava) Mean Number of Real Zeros of a Random Trigonometric Polynomial. II (J E Wilkins, Jr) Orthogonal Polynomials of Many Variables and Degenerated Elliptic Equations (A Yanushauskas) and other papers PREFACE; CONTENTS; ON THE CHARACTERIZATION OF CHEBYSHEV SYSTEMS AND ON CONDITIONS OF THEIR EXTENSION; References; CHARACTERIZATIONS FOR THE EXISTENCE OF A SOLUTION TO THE MOMENT PROBLEM ON A FINITE NUMBER OF INTERVALS; 1. Introduction; 2. Preliminaries and Notations; 2.1. Positive-definite linear functional; 2.2. Orthogonal polynomial theory; 3. Hausdorff Moment Problem; 3.1. Proof of Theorem 2.3; 3.2. A simpler characterization; 4. E is Disconnected; 4.1 . An extension theorem; 4.2. A representation theorem; 4.3. Two interval case; 5. E is a Finite Union of Compact Intervals This volume presents an account of some of the most important work that has been done in various research problems in the theory of polynomials of one and several variables and their applications. It is dedicated to P.L. Chebyshev, a leading Russian mathematician.