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 singularities.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.

Table of Contents:
Spline Functions: Introduction and summary by H. Brunner Radial extensions of vertex data by L. P. Bos and D. Holland The use of splines in the numerical solutions of differential and Volterra integral equations by H. Brunner On best error bounds for deficient splines by F. Dubeau and J. Savoie Optimal error bounds for spline interpolation on a uniform partition by F. Dubeau and J. Savoie Modelization of flexible objects using constrained optimization and B-spline surfaces by J.-P. Dussault and N. Pfister New control polygons for polynomial curves by J. C. Fiorot and P. Jeannin Splines in approximation and differential operators: $(m,\ell,s)$ interpolating-spline by A. Le Mehaute and A. Bouhamidi New families of B-splines on uniform meshes of the plane by P. Sablonniere Theory of Wavelets: Introduction and summary by S. Jaffard Analysis of Hermite-interpolatory subdivision schemes by N. Dyn and D. Levin Some directional microlocal classes defined using wavelet transforms by M. Holschneider Nonseparable biorthogonal wavelet bases of $L^2(\mathbb R^n)$ by A. Karoui and R. Vaillancourt Local bases: Theory and applications by J. Kovacevic and R. Bernardini On the $L^p$-Lipschitz exponents of the scaling functions by K.-S. Lau and M.-F. Ma Robust speech and speaker recognition using instantaneous frequencies and amplitudes obtained with wavelet-derived synchrosqueezing measures by S. Maes Extensions of the Heisenberg group and wavelet analysis in the plane by E. Schulz and K. F. Taylor Wavelets in Physics: Introduction and summary by A. Arneodo Coherent states and quantization by S. Twareque Ali Wavelets in molecular and condensed-matter physics by J.-P. Antoine Wavelets in atomic physics by J.-P. Antoine, Ph. Antoine, and B. Piraux The wavelet $\epsilon$-expansion and Hausdorff dimension by G. Battle Modelling the coupling between small and large scales in the Kuramoto-Sivashinsky equation by J. Elezgaray, G. Berkooz, and P. Holmes Continuous wavelet transform analysis of one-dimensional quantum ground states by C. R. Handy and R. Murenzi Oscillating singularities and fractal functions by A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy Splines and Wavelets in Statistics: Introduction and summary by B. Macgibbon Wavelet estimators for change-point regression models by A. Antoniadis Wavelet thresholding for non (necessarily) Guassian noise: A preliminary report by R. Averkamp and C. Houdre Deslauries-Dubuc: Ten years after by D. L. Donoho and T. P. Y. Yu Some theory for $L$-spline smoothing by J. O. Ramsay and N. Heckman Spectral representation and estimation for locally stationary wavelet processes by R. von Sachs, G. P. Nason, and G. Kroisandt.