Simultaneous approximation by greedy algorithms

We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = ∞ X j=1 cj(f)φj(f), φj(f) ∈ D, j = 1, 2, . . . with the Parseval property: ‖f‖2 = j |cj(f)|. Following the paper of A. Lutoborski and the second author [30] we study analogs of the above expansions for a given finite number of functions f1, . . . , fN with a requirement that the dictionary elements φj of these expansions are the same for all f i, i = 1, . . . , N . We study convergence and rate of convergence of such expansions which we call simultaneous expansions.