Simultaneous approximation by greedy algorithms

We study nonlinear m-term approximation with regard to a redundant dictionary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {D}$\end{document} an expansion into a series \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=\sum_{j=1}^{\infty}c_{j}(f)\varphi_{j}(f),\quad\varphi_{j}(f)\in \mathcal {D},\ j=1,2,\ldots,$$\end{document} with the Parseval property: ‖f‖2=∑j|cj(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 f1,. . .,fN with a requirement that the dictionary elements φj of these expansions are the same for all fi, i=1,. . .,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.