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Spline Functions and the Theory of Wavelets
Edited by: Serge Dubuc, Université de Montréal, QC, Canada, and Gilles Deslauriers, Ecole Polytechnic de Montréal, QC, Canada
A co-publication of the AMS and Centre de Recherches Mathématiques.
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CRM Proceedings & Lecture Notes
1999; 397 pp; softcover
Volume: 18
ISBN-10: 0-8218-0875-3
ISBN-13: 978-0-8218-0875-7
List Price: US$128
Member Price: US$102.40
Order Code: CRMP/18
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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
  • H. Brunner -- Introduction and summary
  • L. P. Bos and D. Holland -- Radial extensions of vertex data
  • H. Brunner -- The use of splines in the numerical solutions of differential and Volterra integral equations
  • F. Dubeau and J. Savoie -- On best error bounds for deficient splines
  • F. Dubeau and J. Savoie -- Optimal error bounds for spline interpolation on a uniform partition
  • J.-P. Dussault and N. Pfister -- Modelization of flexible objects using constrained optimization and B-spline surfaces
  • J. C. Fiorot and P. Jeannin -- New control polygons for polynomial curves
  • A. Le Méhauté and A. Bouhamidi -- Splines in approximation and differential operators: \((m,\ell,s)\) interpolating-spline
  • P. Sablonnière -- New families of B-splines on uniform meshes of the plane
Theory of Wavelets
  • S. Jaffard -- Introduction and summary
  • N. Dyn and D. Levin -- Analysis of Hermite-interpolatory subdivision schemes
  • M. Holschneider -- Some directional microlocal classes defined using wavelet transforms
  • A. Karoui and R. Vaillancourt -- Nonseparable biorthogonal wavelet bases of \(L^2(\mathbb R^n)\)
  • J. Kovačević and R. Bernardini -- Local bases: Theory and applications
  • K.-S. Lau and M.-F. Ma -- On the \(L^p\)-Lipschitz exponents of the scaling functions
  • S. Maes -- Robust speech and speaker recognition using instantaneous frequencies and amplitudes obtained with wavelet-derived synchrosqueezing measures
  • E. Schulz and K. F. Taylor -- Extensions of the Heisenberg group and wavelet analysis in the plane
Wavelets in Physics
  • A. Arneodo -- Introduction and summary
  • S. Twareque Ali -- Coherent states and quantization
  • J.-P. Antoine -- Wavelets in molecular and condensed-matter physics
  • J.-P. Antoine, Ph. Antoine, and B. Piraux -- Wavelets in atomic physics
  • G. Battle -- The wavelet \(\epsilon\)-expansion and Hausdorff dimension
  • J. Elezgaray, G. Berkooz, and P. Holmes -- Modelling the coupling between small and large scales in the Kuramoto-Sivashinsky equation
  • C. R. Handy and R. Murenzi -- Continuous wavelet transform analysis of one-dimensional quantum ground states
  • A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy -- Oscillating singularities and fractal functions
Splines and Wavelets in Statistics
  • B. Macgibbon -- Introduction and summary
  • A. Antoniadis -- Wavelet estimators for change-point regression models
  • R. Averkamp and C. Houdré -- Wavelet thresholding for non (necessarily) Guassian noise: A preliminary report
  • D. L. Donoho and T. P. Y. Yu -- Deslauries-Dubuc: Ten years after
  • J. O. Ramsay and N. Heckman -- Some theory for \(L\)-spline smoothing
  • R. von Sachs, G. P. Nason, and G. Kroisandt -- Spectral representation and estimation for locally stationary wavelet processes
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