In this paper we deal with local Besov spaces of periodic functions of one variable. We characterize these spaces in terms of summability conditions on the coefficients in series expansions of their elements with respect to an orthogonal Schauder basis of trigonometric polynomials. We consider a Schauder basis that was constructed by using ideas of a periodic multiresolution analysis and corres...

Abstract We study approximation classes for adaptive time-stepping finite element methods time-dependent partial differential equations. measure the error in $L_2([0,T)\times \varOmega )$ and consider with discontinuous elements time continuous space, of any degree. As a by-product we define anisotropic Besov spaces Banach-space-valued functions on an interval derive some embeddings, as well Ja...

A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular domains, in particular on the cube. However, he had to exclude essential exceptional values...

Let q ≥ 1 be an integer, S denote the unit sphere embedded in the Euclidean space Rq+1, and μq be its Lebesgue surface measure. We establish upper and lower bounds for sup f∈B p,ρ ∣∣∣∣ ∫ Sq fdμq − M ∑ k=1 wkf(xk) ∣∣∣∣ , xk ∈ S , wk ∈ R, k = 1, · · · ,M, where B p,ρ is the unit ball of a suitable Besov space on the sphere. The upper bounds are obtained for choices of xk and wk that admit exact q...

The multilinear Calderón–Zygmund theory is developed in the setting of RD-spaces, namely, spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón–Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel–Lizorkin spaces in the full range of exponents are among the main...

Let Xn = {x j }j=1 be a set of n points in the d-cube Id := [0, 1]d , and n = {φ j }j=1 a family of n functions on Id . We consider the approximate recovery of functions f on Id from the sampled values f (x1), . . . , f (xn), by the linear sampling algorithm Ln(Xn, n, f ) := ∑nj=1 f (x j )φ j . The error of sampling recovery is measured in the norm of the space Lq(I)-norm or the energy quasi-no...

Abstract This paper considers the radial variation function F (r, t) of an analytic function f(z) on the disc D. We examine F (r, t) when f belongs to a Besov space Apq and look for ways in which F imitates the behaviour of f . Regarded as a function of position (r, t) in D, we show that F obeys a certain integral growth condition which is the real variable analogue of that satisfied by f . We ...

This paper considers coorbit spaces parametrized by mixed, weighted Lebesgue with respect to the quasi-regular representation of semi-direct product Euclidean space and a suitable matrix dilation group. The class groups that we allow, so-called integrably admissible groups, contains yielding an irreducible, square-integrable as proper subclass. obtained scale extends therefore well-studied wave...

In this paper, we demonstrate based on the linear model of [ 1,2] that inverse halftoning is equivalent to the well-studied problem of deconvolution in the presence of colored noise. We propose the use of the simple and elegant wavelet-vaguelette deconvolution (WVD) algorithm to perform the inverse halftoning. Unlike previous wavelet-based algorithms, our method is model-based; hence it is adap...

The aim of the paper is to characterize the trace space of vector-valued Sobolev spaces W p (R , E) , where E is an arbitrary Banach space. In particular, we do not assume that the underlying Banach space E has the UMD property. Vector-valued Sobolev and Besov spaces are widely used in abstract evolution equations, cf. e.g. Amann [1, 2, 4], Veraar and Weis [57] or Denk, Hieber, Prüss, Saal, and...