Exact Constants in Approximation Theory PDF
Preview Exact Constants in Approximation Theory
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS EDITED BY G.-C. ROTA Editorial Board R. S. Doran, J. Goldman, T.-Y. Lam, E. Lutwak Volume 38 Exact constants in approximation theory ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS LUIS A. SANTALO Integral geometry and geometric probability 2 George E. Andrews The theory of partitions 3 Robert J. McEliece The theory of information and coding: a mathematical framework for communication 4 Willard Miller, J r. Symmetry and separation of variables 5 David Ruelle Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics 6 Henryk Mine Permanents 7 Fred S. Roberts Measurement theory with applications to decisionmaking, utility, and the social sciences 8 L. C. Biedenham and J. D. Louck Angular momentum in quantum physics: theory and application 9 L. C. Biedenham and J. D. Louck The Racah-Wigner algebra in quantum theory 10 W. Dollard and Charles N. Friedman Product integration with application to differential equations 11 William B. Jones and W. J. Thran Continued fractions: analytic theory and applications 12 Nathaniel F. G. Martin and James W. England Mathematical theory of entropy 13 George A. Baker, Jr and Peter R. Graves-Morris Pade approximants, Part 1: Basic theory 14 George A. Baker, Jr and Peter R. Graves-Morris Pade approximants, Part II: Extensions and applications 15 E. C. Beltrametti and G. Cassinelli The logic of quantum mechanics 16 G. D. James and A. Kerber The representation theory of the symmetric group 17 M. Lothaire Combinatorics on words 18 H. O. Fattorini The Cauchy problem 19 G. G. Lorentz, K. Jetter, and S. D. Riemenschneider Birkhoff interpolation 20 Rudolf Lidl and Harald Niederreiter Finite fields 21 William T. Tutte Graph theory 22 Julio R. Bastida Field extensions and Galois theory 23 John R. Cannon The one-dimensional heat equation 24 Stan Wagon The Banach-Tarski paradox 25 Arto Salomaa Computation and automata 26 Neil White (ed) Theory ofmatroids 27 N. H. Bingham, C. M. Goldie and J. L. Teugels Regular variation 28 P. P. Petrushev and V. A. Popov Rational approximation of real functions 29 Neil White (ed) Combinatorial geometries 30 M. Pohst and H. Zassenhaus Algorithmic algebraic number theory 31 J. Aczel and J. Dhombres Functional equations containing several variables 32 Marek Kuczma, Bogden Chozewski and Roman Ger Iterative functional equations 33 R. V. Ambartzumian Factorization calculus and geometric probability 34 G. Gripenberg, S.-O. Londen and O. Staffans Volterra integral and functional equations 35 George Gasper and Mizan Rahman Basic hypergeometric series 36 Erik Torgersen Comparison of statistical experiments 37 Arnold Neumaier Interval methods for systems of equations 38 N. Korneichuk Exact constants in approximation theory ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS Exact constants in approximation theory N. KORNEICHUK Institute of Mathematics Ukrainian Academy of Sciences, Kiev Translated from the Russian by K. IVANOV Bulgarian Academy of Sciences, Sofia The right of the University of Cambridge to print a"d sell all manner of books WQS granted by Henry VIII in /534. The Universily has printed and published continuously since 1584. 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QQAA222211..KK66661133 11999911 551111'' . .44----ddcc2200 9900--11444433 CCIIPP IISSBBNN--1133 997788--00--552211--3388223344--22 hhaarrddbbaacckk IISSBBNN--ttoo 00--552211--3388223344--33 hhaarrddbbaacckk TTrraannssffeerrrreedd ttoo ddiiggiittaall pprriinnttiinngg 22000055 CONTENTS –——————————————————————————— Preface IX List of notation Xll 1 Best approximation and duality in extremal problems 1 1.1 Best approximation 1 1.2 Formulation of extremal problems 6 1.3 Duality of extremal problems in linear spaces 10 1.4 Duality in functional spaces 17 1.5 Duality for best approximation of classes of functions 29 Comments 41 Exercises 42 2 Polynomials and spline functions as approximating tools 44 2.1 Polynomials of best approximation 45 2.2 Linear methods for polynomial approximation, Lebesgue constants 51 2.3 Polynomial splines 65 2.4 Spline interpolation 75 2.5 On the existence of perfect splines with prescribed zeros 91 Comments 99 Exercises 100 3 Comparison theorems and inequalities for the norms of functions and their derivatives 101 3.1 Standard splines 101 3.2 Comparison theorems in general cases 107 3.3 Comparison theorems and exact inequalities for differentiable functions 119 3.4 Internal extremal properties of splines 134 3.5 Inequalities for polynomials 144 Comments 152 Other results and exercises 153 v vi Contents 4 Polynomial approximation of classes of functions with bounded rth derivative in Lp 155 4.1 Minimizing the error in the class of -methods 155 4.2 The supremums of the best approximations of classes W by trigonometric polynomials 167 4.3 Approximation by partial sums of Fourier series, their means and analogs 176 4.4 Approximation by algebraic polynomials on an interval 197 Comments 207 Other results and exercises 209 5 Spline approximation of classes of functions with a bounded rth derivative 211 5.1 Inequalities for functions with prescribed zeros 211 5.2 The interpolation error of the periodic splines with minimal defect on classes W 218 5.3 Estimates for spline interpolation on classes W [a,b] 232 5.4 Best approximation by splines of minimal defect 246 Comments 257 Other results and exercises 258 6 Exact constants in Jackson inequalities 260 6.1 Modulus of continuity and general statement of the problems 260 6.2 Jackson inequalities for polynomial approximations 269 6.3 Jackson-type inequalities for spline approximation 283 Comments 295 Other results and exercises 296 7 Approximation of classes of functions determined by modulus of continuity 299 7.1 General facts and -rearrangements 300 7.2 Exact results for the best approximation of classes W Hw, w W H [a,b] 318 7.3 Approximation by linear methods 340 Comments 353 Other results and exercises 354 8 N-widths of functional classes and closely related extremal problems 356 8.1 N-widths of classes of functions with bounded rth derivatives 357 8.2 N-widths of classes defined by the modulus of continuity 377 8.3 Closely related extremal problems 343 Comments 406 Other results and exercises 407 Appendix 410 A1 Holder and Minkowski inequalities and some extremal relations 410 A2 Steklov functions 415 Contents VII A3 Proofs of comparison theorems for l-rearrangements 418 Auxilliary results 418 Proofs of Theorems 7.1.4 and 7.1.5 422 Proof of Theorem 7.1.6 424 References 426 I Monographs 426 A Monographs on approximation theory 426 B Other books 427 II Articles 428 Index of notation 450 Index 452
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