Sets of determination for parabolic functions on a half-space

Jarmila Ranošová

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 3, page 497-513
  • ISSN: 0010-2628

Abstract

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We characterize all subsets M of ℝ n × ℝ + such that sup X ∈ ℝ n × ℝ + u ( X ) = sup X ∈ M u ( X ) for every bounded parabolic function u on ℝ n × 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.

How to cite

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Ranošová, Jarmila. "Sets of determination for parabolic functions on a half-space." Commentationes Mathematicae Universitatis Carolinae 35.3 (1994): 497-513. <http://eudml.org/doc/247599>.

@article{Ranošová1994,
abstract = {We characterize all subsets $M$ of $\mathbb \{R\}^n \times \mathbb \{R\}^+$ such that \[ \sup \limits \_\{X\in \mathbb \{R\}^n \times \mathbb \{R\}^+\}u(X) = \sup \limits \_\{X\in M\}u(X) \] for every bounded parabolic function $u$ on $\mathbb \{R\}^n \times \mathbb \{R\}^+$. 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.},
author = {Ranošová, Jarmila},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {heat equation; parabolic function; Weierstrass kernel; set of determination; decomposition of $L_1(\mathbb \{R\}^n)$; normal distribution; parabolic function; Weierstrass kernels; normal distributions},
language = {eng},
number = {3},
pages = {497-513},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Sets of determination for parabolic functions on a half-space},
url = {http://eudml.org/doc/247599},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Ranošová, Jarmila
TI - Sets of determination for parabolic functions on a half-space
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 3
SP - 497
EP - 513
AB - We characterize all subsets $M$ of $\mathbb {R}^n \times \mathbb {R}^+$ such that \[ \sup \limits _{X\in \mathbb {R}^n \times \mathbb {R}^+}u(X) = \sup \limits _{X\in M}u(X) \] for every bounded parabolic function $u$ on $\mathbb {R}^n \times \mathbb {R}^+$. 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.
LA - eng
KW - heat equation; parabolic function; Weierstrass kernel; set of determination; decomposition of $L_1(\mathbb {R}^n)$; normal distribution; parabolic function; Weierstrass kernels; normal distributions
UR - http://eudml.org/doc/247599
ER -

References

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  1. Aikawa H., Sets of determination for harmonic function in an NTA domains, preprint, 1992. MR1376083
  2. Bonsall F.F., Decomposition of functions as sums of elementary functions, Quart J. Math. Oxford (2) 37 (1986), 129-136. (1986) MR0841422
  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. (1987) Zbl0658.31001MR0908454
  4. Bonsall F.F., Some dual aspects of the Poisson kernel, Proc. Edinburgh Math. Soc. 33 (1990), 207-232. (1990) Zbl0704.31001MR1057750
  5. Doob J.L., Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag New York (1984). (1984) Zbl0549.31001MR0731258
  6. Gardiner S.J., Sets of determination for harmonic function, Trans. Amer. Math. Soc. 338 (1993), 233-243. (1993) MR1100694
  7. Rudin W., Functional Analysis, McGraw-Hill Book Company (1973). (1973) Zbl0253.46001MR0365062
  8. Dudley Ward N.F., Atomic Decompositions of Integrable or Continuous Functions, D.Phil Thesis, University of York, 1991. 

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