Thresholding Greedy Algorithms in Banach Spaces

Resumen

The main objective of this thesis has been to provide a better understanding of the so called Thresholding Greedy Algorithms in Banach spaces. A classical problem in Mathematical Analysis consists in finding representations for a function f as an infinite sum using a basis (or more general, a representation system). Some classical examples of such representations are the Taylor expansions and the Fourier series of functions. On the other hand, a main goal in Approximation Theory is to find good approximations of f in terms of m-terms, that is, fi nite sums supported in a suitable set where the scalars are possibly dif ferent from the original. An m-term algorithm is a deterministic procedure which to each function f and each natural number m, assigns a set and coefficients as we have commented above. In this thesis we have worked with two di fferent m-term algorithms: the Thresholding Greedy Algorithm and the Thresholding Chebyshev Greedy Algorithm. The main results about this algorithms are the study of their convergence respect to some bases, as quasi-greedy, almost-greedy, greedy and semi-greedy bases, and define and study, in each case, the associated Lebesgue-type parameter.