Over the recent years, a new approach for obtaining a succinct approximate representation of ndimensional

Recently, Compressed Sensing (Compressive Sampling) has attracted a lot of attention of both mathematicians and computer scientists. Compressed Sensing refers to a problem of economical recovery of an unknown vector u ∈ R m from the information provided by linear measurements 〈u, ϕj〉, ϕj ∈ R m, j = 1,..., n. The goal is to design an algorithm

This paper concerns special redundant systems, namely, incoherent systems or systems with small coherence parameter. Simple greedy-type algorithms perform well on these systems. For example, known results show that the smaller the coherence parameter, the better the performance of the Orthogonal Greedy Algorithm. These systems with small coherence parameter are also useful in the construction of compressed sensing matrices. Therefore, it is very desirable to build dictionaries with small coherence parameter. We discuss the following problem for both R n and C n: How large can a system with coherence parameter not exceeding a fixed number µ be? We obtain upper and lower bounds for the maximal cardinality of such systems. Although the results herein are either known or simple corollaries of known results, our objective is to demonstrate how fundamental results from different areas of mathematics–linear algebra, probability, and number theory–collaborate on this important approximation theoretic problem.

c ○ Mahdi S. Hosseini 2010I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.

This paper is an attempt to both expound and expand upon, from an approximation theorist’s point of view, some of the theoretical results that have been obtained in the sparse representation (compressed sensing) literature. In particular, we consider in detail ℓm 1-approximation, which is fundamental in the theory of sparse representations, and the connection between the theory of sparse representations and certain n-width concepts. We try to illustrate how the theory of sparse representation leads to new and interesting problems in approximation theory, while the results and techniques of approximation theory can further add to the theory of sparse representations.