Truncation Error on Wavelet Sampling Expansions

We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L ₂(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions.     

Hypercircle inequalities and sampling theory

In this article the well-known hypercircle inequality is extended to the Riesz bases setting. A natural application for this new inequality is given by the estimation of the truncation error in nonorthogonal sampling formulas. Examples including the estimation of the truncation error for wavelet sampling expansions or for nonorthogonal sampling formulas in Paley–Wiener spaces are exhibited.     

On the uniform convergence of wavelet expansions of random processes from Orlicz spaces of random variables. II

We establish conditions under which wavelet expansions of random processes from Orlicz spaces of random variables converge uniformly with probability one on a bounded interval. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 6, pp. 759–775, June, 2008.     

Wavelet Frame Bijectivity on Lebesgue and Hardy Spaces

We prove a sufficient condition for frame-type wavelet series in L ^p , the Hardy space H ¹, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calderón–Zygmund constants of the frame kernel.     

On Approximation with Spline Generated Framelets

We characterize the approximation spaces associated with the best n-term approximation in L_p(R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L_p and classical Besov spaces, and the result coincides with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or ringing artifacts)in the approximants.     

Pointwise convergence of a class of non-orthogonal wavelet expansions

Non-orthogonal wavelet expansions associated with a class of mother wavelets is considered. This class of wavelets comprises mother wavelets that are not necessarily integrable over the whole real line, such as Shannon's wavelet. The pointwise convergence of these wavelet expansions is investigated. It is shown that, unlike other wavelet expansions, the ones under consideration do not necessarily converge pointwise to the functions at points of continuity, unless a more stringent condition, such as bounded variation, is imposed.

     

Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ⁿ, for an affine extension of the Heisenberg group, and on many commutative hypergroups.     

Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ⁿ, for an affine extension of the Heisenberg group, and on many commutative hypergroups.     

Almost Diagonal Matrices and Besov-Type Spaces Based on Wavelet Expansions

This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces \(B^\alpha _{\Psi ,q}(L_p(\Gamma ))\) on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems \(\Psi \) on domains or manifolds \(\Gamma \) which admit a decomposition into smooth patches actually generate the same Besov-type function spaces \(B^\alpha _{\Psi ,q}(L_p(\Gamma ))\), provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces \(b^\alpha _{p,q}(\nabla )\) of Besov type is developed.     

Besov regularity for second order elliptic boundary value problems with variable coefficients

This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝ ^d . The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions. This work has been supported by the Deutsche Forschungsgemeinschaft (Da 360/1-1)     

Hardy's inequalities for Hermite and Laguerre expansions

Hardy's inequalities are proved for higher-dimensional Hermite and special Hermite expansions of functions in Hardy spaces. Inequalities for multiple Laguerre expansions are also deduced.

     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

Fuzzy度量空间扩张型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Characterization of Function Spaces Using Wavelets

It is well known that we can use wavelets to characterize various function spaces, for example, Lebesgue, Sobolev, and Besov spaces, and get equivalent norms with wavelet coefficients. However, we cannot determine whether a function is in these spaces by looking only at the wavelet coefficients since the constant function is orthogonal to all wavelets. In this paper, we close the gap by investigating the convergence of wavelet series.     

Local means,wavelet bases and wavelet isomorphisms in Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces

We consider local means with bounded smoothness for Besov‐Morrey and Triebel‐Lizorkin‐Morrey spaces. Based on those we derive characterizations of these spaces in terms of Daubechies, Meyer, Bernstein (spline) and more general r‐regular (father) wavelets, finally in terms of (biorthogonal) wavelets which can serve as molecules and local means, respectively. Hereby both, local means and wavelet decompositions satisfy natural conditions concerning smoothness and cancellation (moment conditions). Moreover, the given representations by wavelets are unique and yield isomorphisms between the considered function spaces and appropriate sequence spaces of wavelet coefficients. These wavelet representations lead to wavelet bases if, and only if, the function spaces coincide with certain classical Besov‐Triebel‐Lizorkin spaces.     

Fuzzy度量空间型映象的不动点定理

本文给出Ｆｕｚｚｙ度量空间一些扩张型映象的不动点定理,这些结果发展和改进了普通度量空间中相应的结果。     

Necessary and sufficient conditions for expansions of Wilson type

We consider expansions of the type arising from Wilson bases. We characterize such expansions for L^2（R）. As an application, we see that such an expansion must be orthonormal, in contrast to the case of wavelet expansions generated by translations and dilation.     

Gibbs' phenomenon for sampling series and what to do about it

Gibbs' phenomenon occurs for most orthogonal wavelet expansions. It is also shown to occur with many wavelet interpolating series, and a characterization is given. By introducing modifications in such a series, it can be avoided. However, some series that exhibit Gibbs' phenomenon for orthogonal series do not for the associated sampling series.     

Expansions of k-spaces

Simple expansions and expansions by point finite and locally finite collections are studied for particular classes of k-spaces. All such expansions of Fréchet spaces are shown to be Fréchet, and sufficient conditions for the preservation of property P ? {k¹, sequential, k} under simple and locally finite expansions are established.