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...

The problem of representation and analysis of functions defined on manifolds (signals, images, and data in general) is ubiquities in many fields ranging from statistics and cosmology to neuroscience and biology. It is very common to consider input signals as points in a high-dimensional measurement space, however, meaningful structures lay on a manifold embedded in this space. In the last decad...

Frames have turned out to be an essential tool for many applications such as, for example, data transmission, due to their robustness not only against noise but also against losses and due to their freedom in design [4, 6]. Their main advantage lies in the fact that a frame can be designed to be redundant while still providing a reconstruction formula. Since the frame operator Sg = ∑n i=1 〈g, f...

Title of dissertation: Finite Frames and Graph Theoretical Uncertainty Principles Paul J. Koprowski, Doctor of Philosophy, 2015 Dissertation directed by: Dr. John J. Benedetto Department of Mathematics The subject of analytical uncertainty principles is an important field within harmonic analysis, quantum physics, and electrical engineering. We explore uncertainty principles in the context of t...

In this paper, we introduce admissible vectors of covariant representations of a dynamical system which are extensions of the usual ones, and compare them with each other. Also, we give some sufficient conditions for a vector to be admissible vector of a covariant pair of a dynamical system.  In addition, we show the existence of Parseval frames for some special subspaces of $L^2(G)$ related to...

The subject of this article is the duality principle, which, well beyond its stand at the heart of Gabor analysis, is a universal principle in frame theory that gives insight into many phenomena. Its fiber matrix formulation for Gabor systems is the driving principle behind seemingly different results. We show how the classical duality identities, operator representations and constructions for ...

‎In this paper‎, ‎by using the sequence of multipliers‎, ‎we introduce frames with algebraic bounds in Hilbert pro-$ C^* $-modules‎. ‎We investigate the relations between frames and $ ast $-frames‎. ‎Some properties of $ ast $-frames in Hilbert pro-$ C^* $-modules are studied‎. ‎Also‎, ‎we show that there exist two differences between $ ast $-frames in Hilbert pro-$ C^* $-modules and Hilbert $ ...

Abstract In 1946, Dennis Gabor introduced the analytic signal for real‐valued signals f . Here, H is Hilbert transform. This complexification of functions allows an analysis their amplitude and phase information has ever since given well‐interpretable insight into properties over time. The idea been reconsidered with regard to many aspects: examples are dual tree complex wavelet transform, or v...