Abstract
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.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Łysik, G. Mean-value properties of real analytic functions. Arch. Math. 98, 61–70 (2012). https://doi.org/10.1007/s00013-011-0336-0
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DOI: https://doi.org/10.1007/s00013-011-0336-0