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Designed to give a contemporary international survey of research activities in approximation theory and special functions, this book brings together the work of approximation theorists from North America, Western Europe, Asia, Russia, the Ukraine, and several other former Soviet countries. Contents include: results dealing with q-hypergeometric functions, differencehypergeometric functions and basic hypergeometric series with Schur function argument; the theory of orthogonal polynomials and expansions, including generalizations of Szegö type asymptotics and connections with Jacobi matrices; the convergence theory for Padé and Hermite-Padé approximants, with emphasis on techniques from potential theory; material on wavelets and fractals and their relationship to invariant measures and nonlinear approximation; generalizations of de Brange's in equality for univalent functions in a quasi-orthogonal Hilbert space setting; applications of results concerning approximation by entire functions and the problem of analytic continuation; and other topics.



Difference Hypergeometric Functions

The particular solutions for hypergeometric-type difference equations on non-uniform lattices are constructed by using the method of undetermined coefficients. Recently there has been revived interest in the classical theory of special functions. In particular, this theory has been further developed through studying their difference analogs [AWl], [AW2], [NU1], [NSU], [AS1], [AS2], and [S]. Quantum algebras [D], [VS], [KR], [K], which are being developed nowadays, provide a natural basis for the group-theoretic interpretation of these difference special functions. In the present paper we discuss a method of constructing the solutions for difference equations of hypergeometric type on non-uniform lattices [AS2].
N. M. Atakishiyev, S. K. Suslov

Padé Approximants for Some q-Hypergeometric Functions

We show that a large number of explicit formulas for Padé approximants for the ratios of basic hypergeometric functions result from an explicit expression given by Ismail and Rahman for the associated Askey-Wilson polynomials. By specializing this result and using a new transformation for basic hypergeometric series, we are able to recover a result due to Andrews, Goulden and Jackson. We also show how Padé approximants off the main diagonal can be constructed in this latter case.
Mourad E. H. Ismail, Ron Perline, Jet Wimp

Summation Theorems for Basic Hypergeometric Series of Schur Function Argument

In this paper we prove a Ramanujan 1 ψ 1 summation theorem for a Laurent series extension of I.G. Macdonald’s (Schur function) multiple basic hypergeometric series of matrix argument. This result contains as special, limiting cases our Schur function extension of the q-binomial theorem and the Jacobi triple product identity. Just as in the classical case, we write our new q-binomial theorem and 1 ψ 1 summation as elegant special cases of K. Kadell’s and R. Askey’s q-analogs of Selberg’s multiple beta-integral. We also apply our q-binomial theorem and K. Kadell’s Schur function q-analog of Selberg’s beta-integral to derive a Heine transformation and q-Gauss summation theorem for Schur functions.
S. C. Milne

Orthogonal Polynomials, Recurrences, Jacobi Matrices, and Measures

This is a compact bare bone survey of some aspects of orthogonal polynomials addressed primarily to nonspecialists. Special attention is paid to characterization theorems and to spectral properties of Jacobi matrices.
P. Nevai

Szegő Type Asymptotics for Minimal Blaschke Products

Let μ be a positive, finite Borel measure on [0,2π). For 0 <r< 1, 0 <p< ∞, let
$$En,p(d\mu ;r): = _{{B_n}}^{\inf }\left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {{{\left| {{B_n}\left( {r{e^{i\theta }}} \right)} \right|}^p}d\mu \left( \theta \right)} } \right\}1/p,$$
where the infimum is taken over all Blaschke products of ordernhaving zeros in |z| < 1. LetB n * denote a minimal Blaschke product and letG(μ ) denote the geometric mean of the derivative of the absolutely continuous part ofμ. In the first part of the paper we present a self-contained proof of a result due to Parfenov; namelyE n,p ~ r n G(μ′)1/p as n → ∞. In the second part we describe the extension of the classical Szegő function D(z) and prove that B n * (z) ~ z n G(µ′)1/p /D(z)2/p as n → ∞, uniformly on compact subsets of the annulus r < z < 1/r. Some generalizations and applications are also discussed.
A. L. Levin, E. B. Saff

Asymptotics of Hermite-Padé Polynomials

We review results about the asymptotic behavior (in the strong and weak sense) of Hermite-Padé polynomials of type II (also known as German polynomials). The polynomials appear as numerators and denominators of simultaneous rational approximants. The survey begins with general remarks on Hermite-Padé polynomials and a short summary of the state of the theory in this field.
A. I. Aptekarev, Herbert Stahl

On the Rate of Convergence of Padé Approximants of Orthogonal Expansions

A variety of constructions of rational approximations of orthogonal expansions has been discussed in the series of works of 1960–1970 (see [H], [F], [CL], [Gr], and also the monograph of G.A. Baker, Jr. and P. Graves-Morris [BG, Part 2, §1.6]). The greatest interest relates to the definitions of rational approximants which extend the basic definitions (in the sense of Padé-Baker and Frobenius) of the classical Padé approximants of power series to the case of series in orthogonal polynomials. In contrast to the classical case, these definitions lead to substantially different rational approximants of orthogonal expansions. The problems of convergence of the rows of the corresponding Padé tables have been investigated by S. Suetin [S2], [S3], and [Si]. The main results of the present article concern the diagonal Padé approximants of orthogonal expansions. Our purpose is to investigate the rate of convergence of these approximants for Markov type functions.
A. A. Gonchar, E. A. Rakhmanov, S. P. Suetin

Spurious Poles in Diagonal Rational Approximation

Any function f meromorphic in C admits fast rational approximation. That is, if K is a compact set in which f is analytic, there exist rational functions R n of type (n,n),n ≥ 1, such that
$$\mathop {\lim }\limits_{n \to \infty } \left\| {f - {R_n}} \right\|_{{L_\infty }(K)}^{1/n} = 0$$
. More generally, any function f defined on an open set U, and admitting such approximation on a compact KU with positive logarithmic capacity, is said to belong to the Gonchar-Walsh Class on U. We discuss at an introductory, non-technical, level, the problem of spurious poles for diagonal and sectorial sequences of rational approximants to functions in the Gonchar-Walsh class. In particular, we concentrate on some recent positive results on the distribution of poles, and some of their consequences.
D. S. Lubinsky

Expansions for Integrals Relative to Invariant Measures Determined by Contractive Affine Maps

We discuss expansions for integrals to invariant measures of certain stationary Markov chains determined by contractive affine maps. In the homogeneous case, Appell polynomials generated by the Fourier transform of the invariant measure determines the expansion. Some facts about the spectral radius of a stationary subdivision operator and the Lipshitz class of refinable functions are also included.
Charles A. Micchelli

Approximation of Measures by Fractal Generation Techniques

In this paper we discuss the constructive approximation aspects of a measure generating technique that was developed in the last decade by fractal geometers. Measures are generated as stationary distributions of Markov chains that are related to function iteration. The general framework provides a robust machine for constructing parametric families of measures and is, thus, very well suited for measure approximation problems. This formalism — called iterated function system theory — has found application in computer image synthesis and image compression. In fact some of the early work was motivated by a desire to create more realistic images for flight simulators. Other areas of potential application include dynamical systems, signal processing - especially chaotic signals, and quadrature theory.
Stephen Demko

Nonlinear Wavelet Approximation in the Space C(R d )

We discuss the nonlinear approximation of functions from the space C(R d ) by a linear combination of n translated dilates of a fixed function ϕ.
Ronald A. DeVore, Pencho Petrushev, Xiang Ming Yu

Completeness of Systems of Translates and Uniqueness Theorems for Asymptotically Holomorphic Functions

We consider some completeness problems for systems of right and arbitrary translates in certain weighted spaces of functions on the real line. A generalization of the Titchmarsh convolution theorem and a tauberian theorem for quasianalytic Beurling-type algebras are obtained.
A. A. Borichev

Approximation by Entire Functions and Analytic Continuation

This article deals with the application of results from the theory of approximation by entire functions to, classical problems about the analytic continuation of analytic functions given by their Taylor series. Generalizations and completions of well known results due to E. Lindelöf, F. Carlson, and others are obtained.
N. U. Arakelyan

Quasi-Orthogonal Hilbert Space Decompositions and Estimates of Univalent Functions. II

This paper is the second part of a report on an investigation of vectorial Cauchy-Bunyakowskii-Schwarz (CBS) inequality and its applications to estimates of Taylor coefficients of univalent functions. The first part is published in [13] and contains a description of the main general ideas of our approach: CBS inequality for operator measures, quasi-orthogonal (co-isometric) decompositions with respect to complementary metrics, multiplicative averaging of solutions of general evolution equations. The detailed exposition of the theory is contained in [17].
N. K. Nikolskii, V. I. Vasyunin

On the Differential Properties of the Rearrangements of Functions

Let f be a measurable function on a set ER n . In the case E= ∞, we suppose that \( \left| {\left\{ {x \in E:\left| {f\left( x \right)} \right| > y} \right\}} \right| = :{{\lambda }_{f}}\left( y \right) < \infty \) for all y > 0.
V. I. Kolyada

A Class of I.M. Vinogradov’s Series and Its Applications in Harmonic Analysis

The present paper is a survey of the author’s recent research in the one-dimensional trigonometric series of the type
$$\sum\limits_n {\hat f\left( n \right)} {e^{2\pi i\left( {{n^r}{x_r} + \cdots + n{x_1}} \right)}}.$$
K. I. Oskolkov

A Lower Bound for the de Bruijn-Newman Constant Λ. II

A new constructive method is given here for determining lower bounds for the de Bruijn-Newman constant Λ, which is related to the Riemann Hypothesis. This method depends on directly tracking real and nonreal zeros of an entire function F λ(z), where λ < 0, instead of finding, as was previously done, nonreal zeros óf associated Jensen polynomials. We apply this new method to obtain the new lower bound for Λ,-0.385 < Λ, which improves previous published lower bounds of —50 and —5.
T. S. Norfolk, A. Ruttan, R. S. Varga

On the Denseness of Weighted Incomplete Approximations

For a given weight function w(x) on an interval [a, b], we study the generalized Weierstrass problem of determining the class of functions fC[a, b] that are uniform limits of weighted polynomials of the form w n (x)p n (x) 1 , where p n is a polynomial of degree at most n. For a special class of weights, we show that the problem can be solved by knowing the denseness interval of the alternation points for the associated Chebyshev polynomials.
Peter Borwein, E. B. Saff

Asymptotics of Weighted Polynomials

We survey recent developments in the theory of the weighted polynomials w(x) n P n (x), P n P n on a closed set A ⊂ R, with a continuous weight w(x) ≥ 0 on A. Important questions are: Where are the extreme points of the weighted polynomials distributed on A, in particular the alternation points of the weighted Chebyshev polynomials w n C w ,n ? Which continuous functions f on A are approximable by the weighted polynomials? How do the polynomials P n of weighted norm w n P n C (A) = 1 behave outside of A?
This is based on our own work (in particular, in the first two sections) and on work of Mhaskar and Saff, and others.
M. v. Golitschek, G. G. Lorentz, Y. Makovoz
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