Greedy Algorithms and Approximation Properties for Frames in Hilbert Spaces

We study the Threshold Greedy Algorithm (TGA) and Orthogonal Greedy Algorithm (OGA) for frames in Hilbert spaces and obtain various approximation results which are useful in studying the error estimate of the greedy algorithm. We also discuss the relationship between coherence and upper bounds of frames. Lebesgue-type inequalities were proved so as to obtain upper estimates for the errors of the orthogonal greedy algorithm for Riesz frames and frames which are norm bounded below. Various results about the upper estimate of the rate of convergence of the greedy algorithms TGA and OGA with regard to frames are given. Finally, we obtain various relationship among the approximation spaces ℋp, ℒp, and As.