Existence of the best <Emphasis Type="Italic">n</Emphasis>-term approximants for structured dictionaries

We extend the Pizzetti formulas, i.e., expansions of the solid and spherical means of a function in terms of the radius of the ball or sphere, to the case of real analytic functions and to functions of Laplacian growth. We also give characterizations of these functions. As an application we give a characterization of solutions analytic in time of the initial value problem for the heat equation ∂ _t u = Δu in terms of holomorphic properties of the solid and/or spherical means of the initial data.