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Keywords:
heat equation; parabolic function; Weierstrass kernel; set of determination; decomposition of $L_1(\Bbb R^n)$; normal distribution
heat equation; parabolic function; Weierstrass kernel; set of determination; decomposition of $L_1(\Bbb R^n)$; normal distribution
Summary:
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.
References:
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