  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.
