We study the efficient numerical solution of infinite matrix equations Au = f for a matrix A in the Jaffard algebra. These matrices appear naturally via frame discretizations in many applications such as Gabor analysis, sampling theory, and quasi-diagonalization of pseudo-differential operators in the weighted Sjöstrand class. The proposed algorithm has two main features: firstly, it converges ...

We study conditions on a Banach frame that ensures the validity of a reconstruction formula. In particular, we show that any Banach frames for (a subspace of) Lp or Lp,q (1 ≤ p <∞) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulae allows us to prove some James-type results for atomic decompositions: an uncon...

This paper is concerned with the implications of sufficient conditions ensuring that a perturbation of a frame is again a frame. We emphasize how stability of frames is fundamental for numerical applications and we discuss in particular the connection between stability conditions and localization principles for frame atomic decompositions in Banach spaces.

We use several fundamental results which characterize frames for a Hilbert space to give natural generalizations of Hilbert space frames to general Banach spaces. However, we will see that all of these natural generalizations (as well as the currently used generalizations) are equivalent to properties already extensively developed in Banach space theory. We show that the dilation characterizati...

A basic problem of interest in connection with the study of Schauder frames in Banach spaces is that of characterizing those Schauder frames which can essentially be regarded as Schauder bases. In this paper, we study this problem using the notion of a minimal-associated sequence space and a minimal-associated reconstruction operator for Schauder frames. We prove that a Schauder frame is a near...

The intimate connection between the Banach space wavelet reconstruction method on homogeneous spaces with both singular and nonsingular vacuum vectors, and some of well known quantum tomographies, such as: Moyal-representation for a spin, discrete phase space tomography, tomography of a free particle, Homodyne tomography, phase space tomography and SU(1,1) tomography is explained. Also both the...

This paper presents a framework for constructing structured, possibly compactly supported Banach frames and atomic decompositions decomposition spaces. Such space $\def\DecompSp#1#2#3#4{{\mathcal{D}({#1},L_{#4}^{#2},{#3})}}\Deco

The notion of frame in a Banach spaces E via semi-inner product was introduced and studied in [16]. In this paper, we give a characterisation for SIP-II Ed-Bessel sequence to be SIP-II Ed-frame. Also, a necessary and sufficient condition for the finite sum of SIP-II Ed frame for E to be a SIP-II Ed frame for E has been obtained. Further, a sufficient condition for the stability of finite sum of...

A generalized continuous frame is a family of operators on a Hilbert space H which allows reproductions of arbitrary elements of H by continuous superpositions. Generalized continuous frames are natural generalization of continuous and discrete frames in Hilbert spaces which include many recent generalization of frames. In this article,we associate to a generalized continuous frame suitable Ban...

We extend the concept of weaving Hilbert space frames to the Banach space setting. Similar to frames in a Hilbert space, we show that for any two approximate Schauder frames for a Banach space, every weaving is an approximate Schauder frame if and only if there is a uniform constant C ≥ 1 such that every weaving is a C-approximate Schauder frame. We also study weaving Schauder bases, where it i...