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In this paper we review some recent interactions between harmonic analysis and data compression. The story goes back of course to Shannon’s R(D) theory...

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This paper studies the regularity of solutions to boundary value problems for Laplace's equation on Lipschitz domains\Omega in R d and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in L p(\Omega\Gamma are the Besov spaces B ff ø (L ø(\Omega\Gamma16 ø := (ff=d + 1=p) \Gamma1 . Thus, the regularity of the solution in this scale of Besov spaces is investigated with the aim of determining the largest ff for which the solution is in B ff ø (L ø(\Omega\Gamma21 The regularity theorems given in this paper build upon the recent results of Jerison and Kenig [JK]. The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions. Key Words: Besov spaces, elliptic boundary value problems, potential theory, adaptive methods, nonlinear approximation, wavelets AMS Subject classification: primary 35B65, secondary 31B10, 41A46, 46E...

This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis ...

We present a range of mesh-dependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shape-regular (but possibly non-quasiuniform) meshes. These inequalities involve norms of the form kh ff uk W s;p(\Omega\Gamma for positive and negative s and ff, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N , the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results - previously known only for quasiuniform meshes - to the locally refined case. Here we describe applications to: (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.

In this paper, we review several recent developments centering upon the application of multiscale basis methods for the numerical solution of operator equations with special emphasis on complexity questions. In particular, issues like preconditioning, matrix compression, construction of special wavelet bases and adapted error estimators are addressed.

Given a function f 2 L 2 (Q), Q := [0; 1)² and a real number t ? 0, let U(f; t) := inf g2BV(Q) kf \Gamma gk 2 L 2 (I) + t VQ (g); where the infimum is taken over all functions g 2 BV of bounded variation on I. This and related extremal problems arise in several areas of mathematics such as interpolation of operators and statistical estimation, as well as in digital image processing. Techniques for finding minimizers g for U(f; t) based on variational calculus and nonlinear partial differential equations have been put forward by several authors ([DMS], [LOR], [MS], [CL]). The main disadvantage of these approaches is that they are numerically intensive. On the other hand, it is well-known that more elementary methods based on wavelet shrinkage solve related extremal problems, for example, the above problem with BV replaced by the Besov space B 1 1 (L 1 (I)) (see e.g. [CDLL]). However, since BV has no simple description in terms of wavelet coefficients, it is not clear that minimi...

Abstract. A large portion of computation is concerned with approximating a function u. Typically, there are many ways to proceed with such an approximation leading to a variety of algorithms. We address the question of how we should evaluate such algorithms and compare them. In particular, when can we say that a particular algorithm is optimal or near optimal? We shall base our analysis on the approximation error that is achieved with a given (computational or information) budget n. We shall see that the formulation of optimal algorithms depends to a large extent on the context of the problem. For example, numerically approximating the solution to a PDE is different from approximating a signal or image (for the purposes of compression).

The purpose of this paper is to estimate the intensity of some random measure N on a set X by a piecewise constant function on a finite partition of X. Given a (possibly large) family M of candidate partitions, we build a piecewise constant estimator (histogram) on each of them and then use the data to select one estimator in the family. Choosing the square of a Hellinger-type distance as our loss function, we show that each estimator built on a given partition satisfies an analogue of the classical squared bias plus variance risk bound. Moreover, the selection procedure leads to a final estimator satisfying some oracle-type inequality, with, as usual, a possible loss corresponding to the complexity of the family M. When this complexity is not too high, the selected estimator has a risk bounded, up to a universal constant, by the smallest risk bound obtained for the estimators in the family. For suitable choices of the family of partitions, we deduce uniform risk bounds over various classes of intensities. Our approach applies to the estimation of the intensity of an inhomogenous Poisson process, among other counting processes, or the estimation of the mean of a random vector with nonnegative components.