Let H be a finite dimensional (real or complex) Hilbert space and let {ai}i=1 be a non-increasing sequence of positive numbers. Given a finite sequence of vectors F = {fi}pi=1 in H we find necessary and sufficient conditions for the existence of r ∈ N∪{∞} and a Bessel sequence G = {gi}i=1 in H such that F ∪ G is a tight frame for H and ‖gi‖ = ai for 1 ≤ i ≤ r. Moreover, in this case we compute ...

We give a tight scheme for teleporting a quantum state between two parties whose reference frames are misaligned by an action of a finite symmetry group. Unlike previously proposed schemes, ours requires no additional tokens or data to be passed between the participants; the same amount of classical information is transferred as for ordinary quantum teleportation, and the Hilbert space of the e...

Finite tight frames are used widely for many applications. An important problem is to construct finite frames with prescribed norm for each vector in the tight frame. In this paper we provide a fast and simple algorithm for such purpose. Our algorithm employs the Householder transformations. For a finite tight frame consisting m vectors in R or C only O(nm) operations are needed. In addition, w...

Let Ψ = {ψ1, . . . , ψL} ⊂ L2 := L2(−∞,∞) generate a tight affine frame with dilation factor M , where 2 ≤M ∈ Z, and sampling constant b = 1 (for the zeroth scale level). Then for 1 ≤ N ∈ Z, N×oversampling (or oversampling by N) means replacing the sampling constant 1 by 1/N . The Second Oversampling Theorem asserts that N×oversampling of the given tight affine frame generated by Ψ preserves a ...

A frame in an n-dimensional Hilbert space Hn is a possibly redundant collection of vectors {fi}i∈I that span the space. A tight frame is a generalization of an orthonormal basis. A frame {fi}i∈I is said to be scalable if there exist nonnegative scalars {ci}i∈I such that {cifi}i∈I is a tight frame. In this paper we study the combinatorial structure of frames and their decomposition into tight or...

This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly constrained l1-analysis optimization problem, called analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a tight frame provided t...

Introduction to Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Frame Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Tight Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 More Precise Frame Bound...

In this paper we study the fusion frame potential, that is a generalization of the BenedettoFickus (vectorial) frame potential to the finite-dimensional fusion frame setting. The structure of local and global minimizers of this potential is studied, when we restrict the frame potential to suitable sets of fusion frames. These minimizers are related to tight fusion frames as in the classical vec...

Sparsity based regularization methods for image restoration assume that the underlying image has a good sparse approximation under a certain system. Such a system can be a basis, a frame, or a general over-complete dictionary. One widely used class of such systems in image restoration are wavelet tight frames. There have been enduring efforts on seeking wavelet tight frames under which a certai...