Related papers: Frames by Multiplication
In this note, we fix a real invertible $d\times d$ matrix $\mathcal{A}$ and consider $\mathcal{A}\mathbb{Z}^d$ as an index set. For $f\in L^2(\mathbb{R}^d)$, let $\Phi^{\mathcal{A}}_{f}:=\frac{1}{|\det \mathcal{A}|}\sum_{k\in…
A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down…
Recent work in Dynamical Sampling has been centered on characterizing frames obtained by the orbit of a vector under a bounded operator. We prove a necessary and sufficient condition for a pair of bounded commuting operators on a separable…
We characterize Riesz frames and frames with the subframe property and use this to answer most of the questions from the literature concerning these properties and their relationships to the projection methods etc.
Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation…
Optimal sampling of non band-limited functions is an issue of great importance that has attracted considerable attention. We propose to tackle this problem through the use of a frequency warping: First, by a nonlinear shrinking of…
Frames in separable Hilbert spaces gives stable analysis and reconstruction of each vector in the underlying space. In this paper, we study frame conditions for a collection of matrix-valued functions obtained by non-uniform shifts. We give…
We develope a local theory for frames on finite dimensional Hilbert spaces. In particular, a bounded frame on a finite dimensional Hilbert space contains a subset which is a good Riesz basis for a percentage (arbitrarily close to one) of…
In this work, we prove that certain L^2-unbounded transformations of orthogonal wavelet bases generate vaguelets. The L^2-unbounded functions involved in the transformations are assumed to be quasi-homogeneous at high frequencies. We…
In this paper we deal with the connection of frames with the class of Hilbert Schmidt operators. First we give an easy criteria for operators being in this class using frames. It is the equivalent to the criteria using orthonormal bases.…
Weighted average sampling is more practical and numerically more stable than sampling at single points as in the classical Shannon sampling framework. Using the frame theory, one can completely reconstruct a bandlimited function from its…
Frame theory provides a robust method for recovering vectors in a Hilbert space from inner product data, though the associated decomposition formula can be computationally demanding. We relax the frame condition by studying sequences that…
We present a way to construct Parseval frames of piecewise constant functions for $L^2[0,1]$. The construction is similar to the generalized Walsh bases. It is based on iteration of operators that satisfy a Cuntz-type relation, but without…
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…
This work characterizes (dyadic) wavelet frames for $L^2({\mathbb R})$ by means of spectral techniques. These techniques use decomposability properties of the frame operator in spectral representations associated to the dilation operator.…
In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames.…
For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra.…
In this note we investigate the existence of frames of exponentials for $L^2(\Omega)$ in the setting of LCA groups. Our main result shows that sub-multitiling properties of $\Omega \subset \widehat{G}$ with respect to a uniform lattice…
A dual frames multiplier is an operator consisting of analysis, multiplication and synthesis processes, where the analysis and the synthesis are made by two dual frames in a Hilbert space, respectively. In this paper we investigate the…
For every set $S$ of finite measure in $\mathbb{R}$ we construct a discrete set of real frequencies $\Lambda$ such that the exponential system $\{\exp(i\lambda t),\lambda\in\Lambda\}$ is a frame in $L^2(S)$