**CRM Proceedings & Lecture Notes**

Volume: 18;
1999;
397 pp;
Softcover

MSC: Primary 65; 41;
Secondary 42; 94; 81; 62

Print ISBN: 978-0-8218-0875-7

Product Code: CRMP/18

List Price: $128.00

Individual Member Price: $102.40

**Electronic ISBN: 978-1-4704-3932-3
Product Code: CRMP/18.E**

List Price: $128.00

Individual Member Price: $102.40

# Spline Functions and the Theory of Wavelets

Share this page *Edited by *
*Serge Dubuc; Gilles Deslauriers*

A co-publication of the AMS and Centre de Recherches Mathématiques

This work is based on a series of thematic workshops on the theory of wavelets
and the theory of splines. Important applications are included. The volume is
divided into four parts: Spline Functions, Theory of Wavelets, Wavelets in
Physics, and Splines and Wavelets in Statistics.

Part one presents the broad spectrum of current research in the theory and
applications of spline functions. Theory ranges from classical univariate
spline approximation to an abstract framework for multivariate spline
interpolation. Applications include scattered-data interpolation, differential
equations and various techniques in CAGD.

Part two considers two developments in subdivision schemes; one for uniform
regularity and the other for irregular situations. The latter includes
construction of multidimensional wavelet bases and determination of bases with
a given time frequency localization.

In part three, the multifractal formalism is extended to fractal functions
involving oscillating singularites. There is a review of a method of
quantization of classical systems based on the theory of coherent
states. Wavelets are applied in the domains of atomic, molecular and
condensed-matter physics.

In part four, ways in which wavelets can be used to solve important function
estimation problems in statistics are shown. Different wavelet estimators are
proposed in the following distinct cases: functions with discontinuities,
errors that are no longer Gaussian, wavelet estimation with robustness, and
error distribution that is no longer stationary.

Some of the contributions in this volume are current research results not
previously available in monograph form. The volume features many applications
and interesting new theoretical developments. Readers will find powerful
methods for studying irregularities in mathematics, physics, and statistics.

Titles in this series are co-published with the Centre de Recherches Mathématiques.

#### Readership

Graduate students, mathematicians, physicists, and statisticians working in approximation theory, mathematical analysis, image processing, signal analysis, mathematical physics, and function estimation.

# Table of Contents

## Spline Functions and the Theory of Wavelets

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Spline Functions 112
- Introduction and summary 314
- Radial extensions of vertex data 516
- The use of splines in the numerical solutions of differential and Volterra integral equations 1526
- On best error bounds for deficient splines 3344
- Optimal error bounds for spline interpolation on a uniform partition 4152
- Modelization of flexible objects using constrained optimization and B-spline surfaces 5162
- New control polygons for polynomial curves 6576
- Splines in approximation and differential operators: (𝑚,ℓ,𝑠) interpolating-spline 7788
- New families of B-splines on uniform meshes of the plane 89100
- Theory of Wavelets 101112
- Introduction and summary 103114
- Analysis of Hermite-interpolatory subdivision schemes 105116
- Some directional microlocal classes defined using wavelet transforms 115126
- Nonseparable biorthogonal wavelet bases of 𝐿²(ℝⁿ) 135146
- Local bases: Theory and applications 153164
- On the 𝐿^{𝑝}-Lipschitz exponents of the scaling functions 181192
- Robust speech and speaker recognition using instantaneous frequencies and amplitudes obtained with wavelet-derived synchrosqueezing measures 193204
- Extensions of the Heisenberg group and wavelet analysis in the plane 217228
- Wavelets in physics 227238
- Introduction and summary 229240
- Coherent states and quantization 233244
- Wavelets in molecular and condensed-matter physics 245256
- Wavelets in atomic physics 261272
- The wavelet 𝜀-expansion and Hausdorff dimension 277288
- Modelling the coupling between small and large scales in the Kuramoto-Sivashinsky equation 293304
- Continuous wavelet transform analysis of one-dimensional quantum ground states 303314
- Oscillating singularities and fractal functions 315326
- Splines and Wavelets in Statistics 331342
- Introduction and summary 333344
- Wavelet estimators for change-point regression models 335346
- Wavelet thresholding for non (necessarily) Guassian noise: A preliminary report 347358
- Deslauries-Dubuc: Ten years after 355366
- Some theory for 𝐿-spline smoothing 371382
- Spectral representation and estimation for locally stationary wavelet processes 381392
- Back Cover Back Cover1409