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 is an element of H and any dictionary D an expansion into a series f = Sigma(infinity)(j=1) c(j)(f)phi(j)(f), phi(j)(f) is an element of D, j = 1,2,... with the Parseval property: parallel to f parallel to(2) = Sigma(j) | c(j) ( f)|(2). Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f(1),..., f(N) with a requirement that the dictionary elements phi(j) 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.