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Approximate Approximations Mathematical I Surveys I and I Monographs I http://dx.doi.org/10.1090/surv/141 Approximate Approximations Mathematical I Surveys I and I Monographs I Volume 141 I Approximate Approximations I Vladimir Maz'ya I Gunther Schmidt EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 41A30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-141 Library of Congress Cataloging-in-Publication Data Maz'ia, V. G. Approximate approximations / Vladimir Maz'ya, Gunther Schmidt. p. cm. — (Mathematical surveys and monographs ; v. 141) Includes bibliographical references and index. ISBN 978-0-8218-4203-4 (alk. paper) 1. Approximation theory. I. Schmidt, Gunther, 1952- II. Title. QA221.M37 2007 511'.4—dc22 2007060769 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. © The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 Contents Preface xi Chapter 1. Quasi-interpolation 1 1.1. Introduction 1 1.1.1. Exercise for a freshman 1 1.1.2. Simple approximation formula 4 1.2. Further examples 6 1.2.1. Errors of approximate quasi-interpolation 6 1.2.2. A simple application of the approximation formula (1.7) 8 1.2.3. Other basis functions 10 1.2.4. Examples of higher-order quasi-interpolants 13 1.2.5. Examples of multi-dimensional quasi-interpolants 17 Chapter 2. Error estimates for quasi-interpolation 19 2.1. Auxiliary results 19 2.1.1. Function spaces 19 2.1.2. Fourier transform 20 2.1.3. Convolutions 21 2.1.4. Radial functions 22 2.1.5. Multi-dimensional Poisson summation formula 22 2.2. Some properties of quasi-interpolants 23 2.2.1. Young's inequality for semi-discrete convolutions 23 2.2.2. An auxiliary function 25 2.2.3. Representations of the quasi-interpolant Mh,v 27 2.2.4. Relations to continuous convolutions 29 2.2.5. Poisson's summation formula for cr^ 31 2.3. Pointwise error estimates for quasi-interpolation 33 2.3.1. Using the moments of r\ 33 2.3.2. Truncation of summation 35 2.3.3. Local estimate of the quasi-interpolation 36 2.3.4. Holder continuous functions 38 2.3.5. Approximation of derivatives 40 2.4. Lp-estimates of the quasi-interpolation error 41 2.4.1. Formulation of the result 41 2.4.2. Estimation of the remainder term 42 2.4.3. Remark on the best approximation 44 2.4.4. Local Lp-estimates 45 2.5. Notes 45 vi CONTENTS Chapter 3. Various basis functions — examples and constructions 49 3.1. Introduction 49 3.2. Examples 49 3.2.1. One-dimensional examples 49 3.2.2. Examples of multi-variate basis functions 50 3.3. Basis functions for higher-order approximations 50 3.3.1. A general formula 50 3.3.2. Symmetric basis functions 51 3.3.3. Radial basis functions 52 3.3.4. Example 54 3.3.5. Anisotropic Gaussian function 56 3.4. Some other methods 57 3.4.1. Using a collection of different V 58 3.4.2. Derivatives with respect to V 59 3.5. Linear combinations of translates 60 3.5.1. Construction of the generating function (3.30) 60 3.5.2. The case of symmetric r\ 63 3.6. Matrix-valued basis functions 64 3.7. Diminishing the parameter V 65 3.7.1. Perturbation of 77 65 3.7.2. Combinations of translates 67 3.8. Notes 68 Chapter 4. Approximation of integral operators 69 4.1. Introduction 69 4.2. Hilbert transform 71 4.3. Harmonic potentials 74 4.3.1. Action on Gaussians 74 4.3.2. Action on higher-order basis functions 76 4.3.3. Semi-analytic cubature formulas 77 4.3.4. Numerical example 78 4.3.5. Gradient of the harmonic potential 79 4.4. Approximation properties 81 4.4.1. Rough error estimate 81 4.4.2. Quasi-interpolation error in Sobolev spaces of negative order 82 4.4.3. Cubature error for the harmonic potential 85 4.5. Application to boundary integral equation methods 86 4.5.1. Transformation of inhomogeneous problems 87 4.5.2. Error estimates 88 4.6. Notes 91 Chapter 5. Cubature of diffraction, elastic, and hydrodynamic potentials 93 5.1. Diffraction potentials 93 5.1.1. Higher-order formulas 94 5.1.2. Action of the one-dimensional diffraction potential on the Gaussian 94 5.1.3. General form of the convolution with Gaussian 96 5.1.4. Analytic expressions of the diffraction potential 96 5.2. Potentials of advection-diffusion operators 98 CONTENTS vii 5.3. Elastic and hydrodynamic potentials 99 5.4. Two-dimensional potentials 100 5.4.1. Fundamental solutions 100 5.4.2. An auxiliary formula 100 5.4.3. Potentials of the Gaussian 101 5.4.4. Potentials of higher-order generating functions 102 5.4.5. Final form of the elastic and hydrodynamic potentials 105 5.4.6. Using matrix-valued basis functions 106 5.5. Three-dimensional potentials 108 5.5.1. Potentials of the Gaussian 109 5.5.2. Elastic potential of higher-order generating functions 110 5.5.3. Final form of the elastic and hydrodynamic potentials 112 5.6. Notes 113 Chapter 6. Some other cubature problems 115 6.1. Cubature of some pseudodifferential operators 115 6.1.1. Square root of the Laplacian 115 6.1.2. Higher-dimensional singular integrals 116 6.1.3. Biharmonic potential 119 6.2. Approximate solution of non-stationary problems 120 6.2.1. The Cauchy problem for the heat equation 120 6.2.2. The Cauchy problem for the wave equation 123 6.2.3. Vibrations of a plate 125 6.3. Potentials of anisotropic Gaussians 126 6.3.1. Second-order problems 127 6.3.2. Some special cases 128 6.3.3. Elastic and hydrodynamic potentials of anisotropic Gaussians 131 6.3.4. Potentials of orthotropic Gaussians 133 6.4. Potentials of higher-order generating functions for orthotropic Gaussians 134 6.5. Potentials of truncated Gaussians 136 6.5.1. Integral operators over bounded domains 136 6.5.2. Integral operators over the half-space 137 6.5.3. Harmonic potentials 138 6.5.4. Diffraction potentials 139 6.6. Notes 141 Chapter 7. Approximation by Gaussians 143 7.1. Approximation of entire functions 143 7.1.1. Approximants on coarse grids 143 7.1.2. Examples 145 7.2. High-order quasi-interpolation 147 7.2.1. Generating functions 147 7.2.2. Saturation error 148 7.3. Interpolation with Gaussian kernels 151 7.3.1. Lagrangian function 151 7.3.2. Simplification 154 7.3.3. Interpolation error 156 7.3.4. Spectral convergence 157 CONTENTS 7.3.5. Exponential convergence 160 7.4. Orthogonal projection 162 7.5. Notes 163 Chapter 8. Approximate wavelets 165 8.1. Introduction 165 8.2. Approximate refinement equations 168 8.2.1 Approximate refinement equations for r\ 168 8.2.2 Examples of mask functions 170 8.3. Approximate multi-resolution analysis 171 8.3.1 Approximate chain of subspaces 171 8.3.2 Almost orthogonal decomposition 172 8.4. Approximate univariate wavelets 173 8.4.1. Gaussian as scaling function 173 8.4.2 Moments 175 8.4.3 Other examples 177 8.5. Approximate multi-variate wavelet decomposition 181 8.5.1 Projections onto Vo 181 8.5.2 Projections onto WQ 182 8.6. Proof of Theorem 8.6 183 8.6.1 Simplification of G(\) 184 8.6.2 Error of replacing G(X) by g(X) 184 8.6.3 Equations for the Fourier coefficients of 1/g 185 8.6.4 Solution of (8.58) 188 8.6.5 Error of replacing a^ by a^ 189 8.7. Potentials of wavelet basis functions 190 8.7.1 Representation of wavelet basis functions 190 8.7.2, Potentials of anisotropic Gaussians of a complex argument 192 8.7.3, Harmonic potentials of approximate wavelets 193 8.7.4 Diffraction potentials of approximate wavelets 194 8.7.5 Elastic and hydrodynamic potentials of approximate wavelets 194 8.8. Numerical example 195 8.9. Notes 196 Chapter 9. Cubature over bounded domains 197 9.1. Introduction 197 9.2. Simple approach 198 9.3. Application of the approximate refinement equations 200 9.3.1. Approximate factorization of the quasi-interpolant 200 9.3.2. Properties of the mask function fj 201 9.3.3. Convolutions based on the remainder term 202 9.4. Boundary layer quasi-interpolants 203 9.4.1.
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