Series of Faber Polynomials
Series of Faber Polynomials

Author: P.K. Suetin

Publisher: CRC Press

Published: 1998-03-23

Total Pages: 272

ISBN-13: 9789056990589

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Presents some important classical and modern results of the series of Faber polynomials and their applications. Interest in this subject has increased rapidly over the last decade, although the presentation of research has, until now, been confined mainly to journal articles. Applications include theory of functions of complex variables, theory of analytic function approximation, and some aspects of numerical analysis.

Polynomial expansions of analytic functions
Polynomial expansions of analytic functions

Author: Ralph P. Boas

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 85

ISBN-13: 3662251701

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This monograph deals with the expansion properties, in the complex domain, of sets of polynomials which are defined by generating relations. It thus represents a synthesis of two branches of analysis which have been developing almost independently. On the one hand there has grown up a body of results dealing with the more or less formal prop erties of sets of polynomials which possess simple generating relations. Much of this material is summarized in the Bateman compendia (ERDELYI [1], voi. III, chap. 19) and in TRUESDELL [1]. On the other hand, a problem of fundamental interest in classical analysis is to study the representability of an analytic function f(z) as a series ,Lc,. p,. (z), where {p,. } is a prescribed sequence of functions, and the connections between the function f and the coefficients c,. . BIEBERBACH's mono graph Analytische Fortsetzung (Ergebnisse der Mathematik, new series, no. 3) can be regarded as a study of this problem for the special choice p,. (z) =z", and illustrates the depth and detail which such a specializa tion allows. However, the wealth of available information about other sets of polynomials has seldom been put to work in this connection (the application of generating relations to expansion of functions is not even mentioned in the Bateman compendia). At the other extreme, J. M.

Polynomial Expansions of Analytic Functions
Polynomial Expansions of Analytic Functions

Author: Ralph P.Jr. Boas

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 85

ISBN-13: 3642878873

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This monograph deals with the expansion properties, in the complex domain, of sets of polynomials which are defined by generating relations. It thus represents a synthesis of two branches of analysis which have been developing almost independently. On the one hand there has grown up a body of results dealing with the more or less formal prop erties of sets of polynomials which possess simple generating relations. Much of this material is summarized in the Bateman compendia (ERDELYI [1J, vol. III, chap. 19) and in TRUESDELL [1J. On the other hand, a problem of fundamental interest in classical analysis is to study the representability of an analytic function j(z) as a series 2::CnPn(z), where {Pn} is a prescribed sequence of functions, and the connections between the function j and the coefficients en. BIEBERBACH'S mono graph Analytisehe Fortsetzung (Ergebnisse der Mathematik, new series, no. 3) can be regarded as a study of this problem for the special choice Pn (z) = zn, and illustrates the depth and detail which such a specializa tion allows. However, the wealth of available information about other sets of polynomials has seldom been put to work in this connection (the application of generating relations to expansion of functions is not even mentioned in the Bateman compendia). At the other extreme, J. M.

Walsh Equiconvergence of Complex Interpolating Polynomials
Walsh Equiconvergence of Complex Interpolating Polynomials

Author: Amnon Jakimovski

Publisher: Springer Science & Business Media

Published: 2007-05-16

Total Pages: 303

ISBN-13: 1402041756

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This book is a collection of the various old and new results, centered around the following simple and beautiful observation of J.L. Walsh - If a function is analytic in a finite disc, and not in a larger disc, then the difference between the Lagrange interpolant of the function, at the roots of unity, and the partial sums of the Taylor series, about the origin, tends to zero in a larger disc than the radius of convergence of the Taylor series, while each of these operators converges only in the original disc. This book will be particularly useful for researchers in approximation and interpolation theory.

Symmetric Functions and Combinatorial Operators on Polynomials
Symmetric Functions and Combinatorial Operators on Polynomials

Author: Alain Lascoux

Publisher: American Mathematical Soc.

Published:

Total Pages: 282

ISBN-13: 9780821889435

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The theory of symmetric functions is an old topic in mathematics which is used as an algebraic tool in many classical fields. With $\lambda$-rings, one can regard symmetric functions as operators on polynomials and reduce the theory to just a handful of fundamental formulas. One of the main goals of the book is to describe the technique of $\lambda$-rings. The main applications of this technique to the theory of symmetric functions are related to the Euclid algorithm and itsoccurrence in division, continued fractions, Pade approximants, and orthogonal polynomials. Putting the emphasis on the symmetric group instead of symmetric functions, one can extend the theory to non-symmetric polynomials, with Schur functions being replaced by Schubert polynomials. In two independentchapters, the author describes the main properties of these polynomials, following either the approach of Newton and interpolation methods or the method of Cauchy. The last chapter sketches a non-commutative version of symmetric functions, using Young tableaux and the plactic monoid. The book contains numerous exercises clarifying and extending many points of the main text. It will make an excellent supplementary text for a graduate course in combinatorics.

Introduction to Functions of a Complex Variable
Introduction to Functions of a Complex Variable

Author: J. H. Curtiss

Publisher: CRC Press

Published: 2021-07-28

Total Pages: 417

ISBN-13: 1000444953

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This book includes information on elementary general topology, the Cauchy Integral Theorem and concepts of homology and homotopy in their application to the Cauchy theory. It is intended for an introductory course in complex analysis at the first-year graduate and advanced undergraduate level.