 | Michel V. Lectures on constructive approximation: Fourier, spline, and wavelet methods on the real line, the sphere, and the ball. - New York: Birkhäuser: Springer, 2013. - xvi, 326 p.: ill. - (Applied and numerical harmonic analysis). - Bibliogr.: p.307-315. - Ind.: p.317-326. - ISBN 978-0-8176-8402-0
Шифр: (И/В16-М65) 02
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1 Introduction: The Problem to be Solved ....................... 1
Part I Basics
2 Basic Fundamentals: What You Need to Know ................... 13
2.1 Preliminaries .......................................... 13
2.2 Basics of Functional Analysis .......................... 14
2.3 Curves and Surfaces .................................... 26
3 Approximation of Functions on the Real Line ................. 31
3.1 Orthogonal Basis Systems on Intervals .................. 31
3.2 A Brief Introduction to Cubic Splines .................. 56
3.3 An Approximate Identity for R .......................... 63
3.4 The Haar Wavelet ....................................... 69
3.5 A Remark on Higher Dimensions: The Tensor Product
Ansatz ................................................. 80
3.6 Questions for Understanding ............................ 81
Part II Approximation on the Sphere
4 Basic Aspects ............................................... 85
4.1 Some Fundamental Tools ................................. 85
4.2 Questions for Understanding ............................ 96
5 Fourier Analysis ............................................ 97
5.1 Spherical Harmonics .................................... 97
5.2 Fully Normalized Spherical Harmonics .................. 126
5.3 Point Grids ........................................... 130
5.4 Questions for Understanding ........................... 142
6 Spherical Splines .......................................... 145
6.1 Reproducing Kernel Hilbert Spaces ..................... 145
6.2 Spherical Sobolev Spaces .............................. 149
6.1 Spherical Splines ..................................... 101
6.4 Some Remarks .......................................... 177
6.4.1 More General Data .............................. 177
6.4.2 The Spline Matrix .............................. 178
6.4.3 Another Remark on Point Grids .................. 180
6.5 Questions for Understanding ........................... 181
7 Spherical Wavelet Analysis ................................. 183
7.1 Convolutions .......................................... 184
7.2 Scaling Functions ..................................... 190
7.3 Wavelets .............................................. 210
7.4 Numerical Integration on the Sphere (Very Briefly) .... 230
7.5 Questions for Understanding ........................... 237
8 Spherical Slepian Functions ................................ 239
8.1 Spherical Slepian Functions ........................... 239
8.2 Questions for Understanding ........................... 245
Part III Approximation on the 3D Ball
9 Orthonormal Bases .......................................... 249
9.1 Construction and Basic Properties ..................... 249
9.2 Eigenfunctions of Differential Operators .............. 260
9.3 Questions for Understanding ........................... 264
10 Splines .................................................... 265
10.1 Sobolev Spaces ........................................ 266
10.2 Splines on the 3D Ball ................................ 276
10.3 Questions for Understanding ........................... 287
11 Wavelets for Inverse Problems on the 3D Ball ............... 289
11.1 The Problem ........................................... 289
11.2 Convolutions .......................................... 291
11.3 Scaling Functions ..................................... 293
11.4 Wavelets .............................................. 297
11.5 Questions for Understanding ........................... 302
12 The Regularized Functional Matching Pursuit ................ 303
12.1 The Idea .............................................. 303
12.2 Questions for Understanding ........................... 305
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