Rate of convergence of Thresholding Greedy Algorithms

The rate of convergence of the classical Thresholding Greedy Algorithm with respect to bases is studied in this paper. We bound the error of approximation by the product of both norms -- the norm of $f$ and the $A_1$-norm of $f$. We obtain some results for greedy bases, unconditional bases, and quasi-greedy bases. In particular, we prove that our bounds for the trigonometric basis and for the Haar basis are optimal.