Rational Approximation of Real Functions
Cambridge University Press, Mar 3, 2011 - Mathematics - 384 pages
Originally published in 1987, this book is devoted to the approximation of real functions by real rational functions. These are, in many ways, a more convenient tool than polynomials, and interest in them was growing, especially since D. Newman's work in the mid-sixties. The authors aim at presenting the basic achievements of the subject and, for completeness, also discuss some topics from complex rational approximation. Certain classical and modern results from linear approximation theory and spline approximation are also included for comparative purposes. This book will be of value to anyone with an interest in approximation theory and numerical analysis.
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Qualitative theory of the best rational approximation
Some classical results in the linear theory
Approximation of some important functions
Converse theorems for rational approximation
Spline approximation and Besov spaces
Relations between rational and spline approximations
Approximation with respect to Hausdorff distance
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A.A. Gonchar absolute constant algebraic polynomial alternate signs approximation in Lp approximation of functions Besov spaces best approximation best rational approximation best uniform approximation bounded functions bounded variation chapter characterization complemented graph Consequently converse theorems convex functions corollary defined definition denote direct and converse Dolzenko exact exists a rational finite following lemma function f function of order functions of bounded give Hardy-Littlewood maximal function Hausdorff distance Hence Holder's inequality implies integral interval lemma Let f Let us consider Let us set metric projection modulus of continuity modulus of smoothness natural number normed o-effect obtain Obviously order of approximation Pade approximants Peetre Pekarskii Petrushev points poles Proof of theorem properties prove theorem quasi-norm rational function rational uniform approximation Remark Remez algorithm satisfies section 7.1 Sendov signx spline approximation subsection subspace Suppose theorem 8.1 theory trigonometric polynomial uniform rational approximation zeros