Jarmila Rano\v sov\'a
Sets of determination for parabolic functions on a half-space

Comment.Math.Univ.Carolinae 35,3 (1994) 497-513.

Abstract:We characterize all subsets $M$ of $\Bbb R^n \times \Bbb R^+$ such that $$ \sup \limits _{X\in \Bbb R^n \times \Bbb R^+}u(X) = \sup \limits _{X\in M}u(X) $$ for every bounded parabolic function $u$ on $\Bbb R^n \times \Bbb R^+$. The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of $M$ is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Keywords: heat equation, parabolic function, Weierstrass kernel, set of determination, decomposition of $L_1(\Bbb R^n)$, normal distribution
AMS Subject Classification: 35K05, 35K15, 31B10, 60Exx

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