Multivariate Polysplines: Applications to Numerical and Wavelet Analysis / Edition 1

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More About This Textbook

Overview

Multivariate polysplines are a new mathematical technique that has arisen from a synthesis of approximation theory and the theory of partial differential equations. It is an invaluable means to interpolate practical data with smooth functions.
Multivariate polysplines have applications in the design of surfaces and "smoothing" that are essential in computer aided geometric design (CAGD and CAD/CAM systems), geophysics, magnetism, geodesy, geography, wavelet analysis and signal and image processing. In many cases involving practical data in these areas, polysplines are proving more effective than well-established methods, such as kKriging, radial basis functions, thin plate splines and minimum curvature.

Part 1 assumes no special knowledge of partial differential equations and is intended as a graduate level introduction to the topic
Part 2 develops the theory of cardinal Polysplines, which is a natural generalization of Schoenberg's beautiful one-dimensional theory of cardinal splines.
Part 3 constructs a wavelet analysis using cardinal Polysplines. The results parallel those found by Chui for the one-dimensional case.
Part 4 considers the ultimate generalization of Polysplines - on manifolds, for a wide class of higher-order elliptic operators and satisfying a Holladay variational property.

Audience: Applied and pure mathematicians, computer scientists and researchers and engineers in signal and image processing, CAGD and CAD/CAM systems, geophysics, geography, magnetism and related disciplines.

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Product Details

  • ISBN-13: 9780124224902
  • Publisher: Elsevier Science
  • Publication date: 7/4/2001
  • Edition number: 1
  • Pages: 498
  • Product dimensions: 1.19 (w) x 6.14 (h) x 9.21 (d)

Meet the Author

Ognyan Kounchev received his M.S. in partial differential equations from Sofia University, Bulgaria and his Ph.D. in optimal control of partial differential equations and numerical methods from the University of Belarus, Minsk. He was awarded a grant from the Volkswagen Foundation (1996-1999) for studying the applications of partial differential equations in approximation and spline theory. Currently, Dr Kounchev is a Fulbright Scholar at the University of Wisconsin-Madison where he works in the Wavelet Ideal Data Representation Center in the Department of Computer Sciences.

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Table of Contents

Preface
1 Introduction 1
Pt. I Introduction to polysplines 15
2 One-dimensional linear and cubic splines 19
3 The two-dimensional case: data and smoothness concepts 29
4 The objects concept: harmonic and polyharmonic functions in rectangular domains in R[superscript 2] 39
5 Polysplines on strips in R[superscript 2] 57
6 Application of polysplines to magnetism and CAGD 67
7 The objects concept: harmonic and polyharmonic functions in annuli in R[superscript 2] 77
8 Polysplines and annuli in R[superscript 2] 101
9 Polysplines on strips and annuli in R[superscript n] 117
10 Compendium on spherical harmonics and polyharmonic functions 129
11 Appendix on Chebyshev splines 187
12 Appendix on Fourier series and Fourier transform 209
Bibliography to Part I 213
Pt. II Cardinal polysplines in R[superscript n] 217
13 Cardinal L-splines according to Micchelli 221
14 Riesz bounds for the cardinal L-splines Q[subscript Z+1] 267
15 Cardinal interpolation polysplines on annuli 287
Bibliography to Part II 307
Pt. III Wavelet analysis 309
16 Chui's cardinal spline wavelet analysis 313
17 Cardinal L-spline wavelet analysis 325
18 Polyharmonic wavelet analysis: scaling and rotationally invariant spaces 371
Bibliography to Part III 395
Pt. IV Polysplines for general interfaces 397
19 Heuristic arguments 399
20 Definition of polysplines and uniqueness for general interfaces 409
21 A priori estimates and Fredholm operators 429
22 Existence and convergence of polysplines 445
23 Appendix on elliptic boundary value problems in Sobolev and Holder spaces 461
24 Afterword 485
Bibliography to Part IV 487
Index 491
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