Related papers: A Guide to Localized Frames and Applications to Ga…
We introduce a localization concept for operator-valued frames, where the quality of localization is measured by the associated operator-valued Gram matrix belonging to some suitable Banach algebra. We prove that intrinsic localization of…
To be able to solve operator equations numerically a discretization of those operators is necessary. In the Galerkin approach bases are used to achieve discretized versions of operators. In a more general set-up, frames can be used to…
We study mapping properties of two-dimensional linear integral operators in some weighted spaces with special kernels. The considered spaces are certain variant of Sobolev--Slobodetskii spaces and their generalizations related to Banach…
This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that…
We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers,…
We consider linear bounded operators acting in Banach spaces with a basis, such operators can be represented by an infinite matrix. We prove that for an invertible operator there exists a sequence of invertible finite-dimensional operators…
In this paper, we propose to define the concept of family of local atoms and then we generalize this concept to the atomic system for operator in Banach spaces by using semi-inner product. We also give a characterization of atomic systems…
In solving scientific, engineering or pure mathematical problems one is often faced with a need to approximate the function of a given class by the linear combination of a preferably small number of functions that are localised one way or…
Matrix type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such…
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices,…
In this paper, we investigate whether the tensor product of two frames, each individually localised with respect to a spectral matrix algebra, is also localised with respect to a suitably chosen tensor product algebra. We provide a partial…
We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and…
We present a general framework of localized operators, i.e., operators whose matrix coefficients with respect to the Gabor frame are concentrated on the diagonal. We show that localized operators are bounded between modulation spaces, and…
In this paper we present a rare combination of abstract results on the spectral properties of slanted matrices and some of their very specific applications to frame theory and sampling problems. We show that for a large class of slanted…
The Lizorkin space is well-suited for studying various operators; e.g., fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. However, we can show that…
The purpose of this paper is to study an implicit scheme for a representation of nonexpansive mappings on a closed convex subset of a smooth and uniformly convex Banach space with respect to a left regular sequence of means defined on an…
Frame multipliers are an abstract version of Toeplitz operators in frame theory and consist of a composition of a multiplication operator with the analysis and synthesis operators. Whereas the boundedness properties of frame multipliers on…
This paper studies boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships…
The purpose of the paper is to analyze frames $\{f_k\}_{k\in \mathbf Z}$ having the form $\{T^kf_0\}_{k\in\mathbf Z}$ for some linear operator $T: \mbox{span} \{f_k\}_{k\in \mathbf Z} \to \mbox{span}\{f_k\}_{k\in \mathbf Z}$. A key result…
A development and an application of the localized function method based on the Galerkin method applying a set of Sine functions to approximate the unknown mode fields of the localized modes propagating along the photonic crystal fibres is…