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We propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 { + v0, where [u0,v0] is the minimizer of a J-functional, J(f, λ0; X, Y) = infu+v=f ‖u‖X + λ0‖v ‖ p} Y. Such minimizers are standard tools for image manipulations

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We establish learning rates up to the order of n −1 for support vector machines with hinge loss (L1-SVMs) and nontrivial distributions. For the stochastic analysis of these algorithms we use recently developed concepts such as Tsybakov’s noise assumption and local Rademacher averages. Furthermore we introduce a new geometric noise condition for distributions that is used to bound the approximation error of Gaussian kernels in terms of their widths. 1

This is an update of, and a supplement to, the author’s earlier survey paper [18] on basic properties of strong mixing conditions. That paper appeared in 1986 in a book containing survey papers on various types of dependence conditions and the limit theory under them. The survey here will include part (but not all) of the material in [18], and will also describe some relevant material that was not in that paper, especially some new discoveries and developments that have occurred since that paper was published. (Much of the new material described here involves “interlaced ” strong mixing conditions, in which the index sets are not restricted to “past ” and “future.”) At various places in this survey, open problems will be posed. There is a large literature on basic properties of strong mixing conditions. A survey such as this cannot do full justice to it. Here are a few references on important topics not covered in this survey. For the approximation of mixing sequences by martingale differences, see e.g. the book by Hall and Heyde [80]. For the direct approximation of mixing random variables by independent ones,

We find upper and lower bounds for Pr ( P x n t), where x 1 , x 2 ; : : : are real numbers. We express the answer in terms of the K-interpolation norm from the theory of interpolation of Banach spaces.

original: 10.1007/s00365-005-0621-x publication is available at springerlink.com with DOI: 10.1007/s00365-005-0621-x

We study nonlinear n-term approximation in Lp(R2) (0 < p < ∞) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of R2 which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces generated by multilevel nested triangulations. We call them B-spaces because they can be viewed as generalizations of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates for n-term piecewise polynomial approximation and consequently characterize the corresponding approximation spaces by interpolation. We also develop methods for n-term piecewise polynomial approximation which capture the rates of the best approximation.

. This paper is the continuation of [17]. We investigate mapping and spectral properties of pseudodifferential operators of type \Psi 1;fl with 2 IR and 0 fl 1 in the weighted function spaces B s p;q (IR n ; w(x)) and F s p;q (IR n ; w(x)) treated in [17]. Furthermore we study the distribution of eigenvalues and the behaviour of corresponding root spaces for degenerate pseudodifferential operators preferably of type b2 (x)b(x; D)b1(x), where b1 (x) and b2(x) are appropriate functions and b(x; D) 2 \Psi 1;fl . Finally, on the basis of the Birman-Schwinger principle, we deal with the "negative spectrum" (bound states) of related symmetric operators in L2 . Math. Subject Classification: 46E35, 47G30, 35S05 1. Introduction The spaces B s p;q and F s p;q with s 2 IR; 0 ! p 1 (p ! 1 for the F - spaces) and 0 ! q 1 on IR n cover many well-known classical spaces such as (fractional) Sobolev spaces, H older-Zygmund spaces, Besov spaces and (inhomogeneous) Hardy spaces. In [1...

In this paper we give two different variational characterizations for the eigenvalues of H + V where H denotes the free Dirac operator and V is a scalar potential. The first one is a min-max involving a Rayleigh quotient. The second one consists in minimizing an appropriate nonlinear functional. Both methods can be applied to potentials which have singularities as strong as the Coulomb potential.