We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...

In this paper we establish an optimal Lorentz estimate for the Riesz potential in L1 regime setting of a stratified group G: Let Q?2 be homogeneous dimension G and I? denote order ? on G. Then, every ??(0,Q), there exists constant C=C(?,Q)>0 such that (0.1)?I?f?LQ?(Q??),1(G)?C?XI1f?L1(G)for all f?Cc?(G) XI1f?L1(G), where X denotes horizontal gradient.

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L 2 (G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Ries...

Let G be the split solvable Lie group acting simply transitively on a Siegel domain D. We consider irreducible unitary representations of G realized on Hilbert spaces of holomorphic functions on D. We determine all such Hilbert spaces by connecting them with positive Riesz distributions on the dual cone and describe them through the Fourier-Laplace transform. Moreover we classify the representa...

Image quality assessment is an important topic in the field of digital image processing. In this study, a full-reference image quality assessment method called Riesz transform and Visual contrast sensitivity-based feature SIMilarity index (RVSIM) is proposed. More precisely, a Log-Gabor filter is first used to decompose reference and distorted images, and Riesz transform is performed on the dec...

We derive an adaptive solver for random elliptic boundary value problems, using techniques from adaptive wavelet methods. Substituting wavelets by polynomials of the random parameters leads to a modular solver for the parameter dependence, which combines with any discretization on the spatial domain. We show optimality properties of this solver, and present numerical computations. Introduction ...

Abstract. We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel’s inequality, a...