Studies in Spline Functions and Approximation Theory PDF
Preview Studies in Spline Functions and Approximation Theory
STUDIES IN SPLINE FUNCTIONS AND APPROXIMATION THEORY ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION STUDIES IN SPLINE FUNCTIONS AND APPROXIMATION THEORY Samuel Karlin Department of Mathematics Weizmann Institute of Science Rehovot, Israel and Department of Mathematics Stanford University Stanford, California Charles A. Micchelli IBM Corporation T.J. Watson Research Center Yorktown Heights, New York Allan Pinkus Department of Mathematics Weizmann Institute of Science Rehovot, Israel I. J. Schoenberg Mathematics Research Center University of Wisconsin Madison, Wisconsin ACADEMIC PRESS, INC. New York San Francisco London 1976 A Subsidiary of Harcourt Brace Jovanovich, Publishers Copyright © 1976, by Academic Press, Inc. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NW1 Library of Congress Cataloging in Publication Data Main entry under title: Studies in spline functions and approximation theory. Bibliography: p. Includes index. 1. Spline theory. 2. Approximation theory. I. Karlin, Samuel, (dated) QA224.S85 51l'.4 76-3453 ISBN 0-12-398565-X PRINTED IN THE UNITED STATES OF AMERICA CONTENTS PREFACE XI ABSTRACTS 1 PART I. BEST APPROXIMATIONS, OPTIMAL QUADRATURE AND MONOSPLINES t On a Class of Best Non-linear Approximation Problems and Extended Monosplines 19 Samuel Karlin §1 Formulation and description of main results. 19 2 Bounds on the number of zeros of extended monosplines. 27 3 The fundamental theorem of algebra for extended monosplines. 36 4 The improvement theorem. 37 5 Existence of minimizing extended splines. 44 6 Proof of the characterization of the extremum, as described 47 in Theorem 1.3 for 1 < p < 00. 7 Characterization and uniqueness in the case of p = 00 51 in Theorem 1.3. 8 The total positivity nature of the Bergman and Szegô reproducing 55 kernels. 9 An equivalent formulation of the best non-linear approximation 59 problem (1.4) in L2. A Global Improvement Theorem for Polynomial Monosplines 67 Samuel Karlin § 1 Statement of theorem and ramifications. 67 2 Some preliminaries. 72 3 Proof of Theorem 3. 74 Applications of Representation Theorems to Problems of Chebyshev Approximation with Constraints 83 Allan Pinkus §1 Introduction. 83 2 Proof of Theorem 2 and extensions of Theorem 1. 87 V CONTENTS 3 Applications of Theorem 2. 89 4 Representation theorems with interpolation, and interpolatory approximation. 94 5 Representation theorems with boundary constraints. 98 6 Applications of representation theorems with boundary conditions. 103 Gaussian Quadrature Formulae with Multiple Nodes 113 Samuel Karlin and Allan Pinkus § 1 Formulation and statement of main results. 113 2 Proof of Theorem 1 and ramifications. 121 3 Extensions and remarks. 137 An Extremal Property of Multiple Gaussian Nodes 143 Samuel Karlin and Allan Pinkus § 1 Formulation and statement of results. 143 2 Preliminaries and some determinantal identities. 149 3 Proof of extremal property (Theorem D). 154 PART II. CARDINAL SPLINES AND RELATED MATTERS Oscillation Matrices and Cardinal Spline Interpolation 163 Charles A. Micchelli §1 Introduction. 163 2 Quasi-Hermite cardinal interpolation. 168 3 Cardinal <£-splines. 184 4 An eigenvalue problem. 189 Cardinal JC-Splines 203 Charles A. Micchelli §1 Introduction. 203 2 An eigenvalue problem. 205 3 Some special £ -splines. 220 4 Convergence of JC-splines. 224 5 Some applications of the Euler and Bernoulli £-spline. 228 6 Polynomial spline interpolation on a geometric mesh. 241 On Micchelli’s Theory of Cardinal ¿-Splines 251 I.J. Schoenberg §1 Introduction. 251 2 The class S (£,r?) of cardinal JC-splines. 253 3 The B-splines. 255 vi CONTENTS 4 The behavior of the roots of the equation (3.4). 257 5 A proof of Theorem 2. 261 6 Micchelli’s cardinal interpolation problem by elements of S(£,t?). 266 7 Proof of sufficiency in Theorem 4. 269 8 A few examples. 273 On The Remainders and the Convergence of Cardinal Spline Interpolation for Almost Periodic Functions 277 /./. Schoenberg § 1 Introduction. 277 2 The kernel of the remainder and some of its properties. 278 3 Further properties of the kernel K2 j (x,t). 283 4 The cardinal spline interpolation formula with remainder. 286 5 A few special choices of f(x). 288 6 Applications. 291 7 Definitions and known results. 295 8 Katznelson’s definition and lemma. 297 9 Two applications of Katznelson’s lemma. 298 10 A conjecture. 302 PART III. INTERPOLATION WITH SPLINES Interpolation by Splines with Mixed Boundary Conditions 305 Samuel Karlin and Allan Pinkus § 1 Introduction. 305 2 Proof of Theorem 1. 309 3 Examples of boundary conditions satisfying Postulate J. 314 4 Total positivity properties of Green’s function for mixed 318 boundary conditions. Divided Differences and Other Non-linear Existence Problems at Extremal Points 327 Samuel Karlin and Allan Pinkus § 1 Preliminaries and statement of main results. 327 2 Polynomials with prescribed kth order divided differences: Proof of Theorem 1. 330 3 Extensions of Theorem 1 to various classes of spline functions and Chebyshev systems. 339 4 A general formulation and applications. 345 5 Open problems. 349 vii CONTENTS PART IV. GENERALIZED LANDAU AND MARKOV TYPE INEQUALITIES AND GENERALIZED PERFECT SPLINES. Notes on Spline Function VI. Extremum Problems of the Landau-Type for the Differential Operators D2 ± 1 353 /./. Schoenberg §0 Introduction. 353 1 The differential operator D2 + 1. 354 2 The differential operator D2 - 1. 359 3 Weak extremum functions in Theorem 3. 364 Oscillatory Perfect Splines and Related Extremal Problems 371 Samuel Karlin § 1 Introduction and statement of main results. 371 2 The construction of certain generalized perfect splines with special oscillatory properties. 385 3 Some special oscillatory perfect splines. 394 4 Uniqueness criteria, proof of Theorem 4.1 and related matters. 396 5 A special one parameter family of equi-oscillation perfect splines. 403 6 Results on equi-oscillating perfect splines with variable interval length. 412 7 Some symmetry considerations. 417 8 Some extremal properties of the equi-oscillating perfect splines. 418 9 Perfect L-splines. 423 10 Equi-oscillating perfect L-splines on [ 0,°°) and on (-°°, °°) with an infinite number of knots. 426 11 Perfect L-splines for n-th order differential operators with constant coefficients on the half line and sharp Landau Kolmogorov type inequalities. 432 12 Equi-oscillating perfect L-splines for the full line where L is a differential operator with constant coefficients. 443 13 Landau Kolmogorov type inequalities for certain convolution 453 operators. Generalized Markov Bernstein Type Inequalities for Spline Functions 461 Samuel Karlin § 1 Introduction and statement of main results. 461 2 Markov inequalities for perfect splines: Proof of Theorem 1. 469 3 Markov inequalities for one-sided and two-sided “Cardinal splines”. 476 Viii CONTENTS Some One-sided Numerical Differentiation Formulae and Applications 485 Samuel Karlin §0 Introduction. 485 1 The finite interval case. 486 2 The one-sided infinite interval case. 494 3 Remarks. 499 ix
The list of books you might like
