Let {xn} be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {xn} which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {xn} can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is ...

The purpose of this paper is to study the arithmetic function f : Z+ → Q ∗ + defined by f(2l) = l (∀k, l ∈ N, l odd). We have, for example, f(1) = 1, f(2) = 1, f(3) = 3, f(12) = 1 3 , f(40) = 1 25 , . . . , so it is clear that f(n) is not always an integer. However, we will show in what follows that f satisfies the property that the product of the f(r) for 1 ≤ r ≤ n is always an integer, and it...

in this paper we introduce and study besselian $g$-frames. we show that the kernel of associated synthesis operator for a besselian $g$-frame is finite dimensional. we also introduce $alpha$-dual of a $g$-frame and we get some results when we use the hilbert-schmidt norm for the members of a $g$-frame in a finite dimensional hilbert space.

Let p be a prime, Cp the completion of an algebraic closure of the p-adicnumbers Qp and K a finite extension of Qp contained in Cp. Let v be the valuation on Cp such that v(p) = 1 and let | | be the absolute value on Cp such that |x| = p for x ∈ Cp. Suppose N is a positive integer prime to p. Let X1(Np) denote the modular curve over K which represents elliptic curves with Γ1(Np)-structure and l...

A common method to compute sparse solutions in compressed sensing is based on `minimization. In this paper we consider `-minimization for arbitrary 0 < p ≤ 1. More precisely, we prove stability estimates for solutions of the analysis formulation of the `minimization problem for 0 < p ≤ 1 from noisy measurements. Furthermore, our focus lies in arbitrary frames that are not necessarily tight or P...

In this paper we seek to describe the structure of self-dual torsion-free LCA groups. We first present a proof of the structure theorem of self-dual torsion-free metric LCA groups. Then we generalize the structure theorem to a larger class of selfdual torsion-free LCA groups. We also give a characterization of torsion-free divisible LCA groups. Consequently, a complete classification of self-du...

We consider a generator F 1⁄4 ðf1; y ;fNÞ for either a multi-frame or a super-frame generated under the action of a projective unitary representation for a discrete countable group. Examples of such frames include Gabor multi-frames, Gabor super-frames and frames for shift-invariant subspaces. We show that there exists a unique normalized tight multi-frame (resp. super-frame) generator C 1⁄4 ðc...