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 = ∑j=1∞cj(f) ψj(f), ψj(f)∈ D, j = 1, 2,..., with the Parseval property: ∥f∥² = ∑_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 φ_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.