Previous |  Up |  Next


heat equation; parabolic function; Weierstrass kernel; set of determination; decomposition of $L_1(\Bbb R^n)$; normal distribution
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.
[1] Aikawa H.: Sets of determination for harmonic function in an NTA domains. preprint, 1992. MR 1376083
[2] Bonsall F.F.: Decomposition of functions as sums of elementary functions. Quart J. Math. Oxford (2) 37 (1986), 129-136. MR 0841422
[3] Bonsall F.F.: Domination of the supremum of a bounded harmonic function by its supremum over a countable subset. Proc. Edinburgh Math. Soc. 30 (1987), 441-477. MR 0908454 | Zbl 0658.31001
[4] Bonsall F.F.: Some dual aspects of the Poisson kernel. Proc. Edinburgh Math. Soc. 33 (1990), 207-232. MR 1057750 | Zbl 0704.31001
[5] Doob J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer-Verlag New York (1984). MR 0731258 | Zbl 0549.31001
[6] Gardiner S.J.: Sets of determination for harmonic function. Trans. Amer. Math. Soc. 338 (1993), 233-243. MR 1100694
[7] Rudin W.: Functional Analysis. McGraw-Hill Book Company (1973). MR 0365062 | Zbl 0253.46001
[8] Dudley Ward N.F.: Atomic Decompositions of Integrable or Continuous Functions. D.Phil Thesis, University of York, 1991.
Partner of
EuDML logo