  It is a survey on recent results in constructive sparse approximation. Three
directions are discussed here: (1) Lebesgue-type inequalities for greedy
algorithms with respect to a special class of dictionaries, (2) constructive
sparse approximation with respect to the trigonometric system, (3) sparse
approximation with respect to dictionaries with tensor product structure. In
all three cases constructive ways are provided for sparse approximation. The
technique used is based on fundamental results from the theory of greedy
approximation. In particular, results in the direction (1) are based on deep
methods developed recently in compressed sensing. We present some of these
results with detailed proofs.
