Uniform approximation of harmonic functions

G. F. Vincent-Smith

Annales de l'institut Fourier (1969)

  • Volume: 19, Issue: 2, page 339-353
  • ISSN: 0373-0956

Abstract

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Let ω be an open relatively compact weakly determining subset of a locally compact harmonic space in the axiomatic of Boboc-Constantinescu-Cornea. If f is continuous on ω ‾ and harmonic in ω the f may be uniformly approximated on ω ‾ to within ϵ by a function harmonic in an open set containing ω ‾ . The proof uses an extension of the Weierstrass-Stone theorem to geometric simplexes.

How to cite

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Vincent-Smith, G. F.. "Uniform approximation of harmonic functions." Annales de l'institut Fourier 19.2 (1969): 339-353. <http://eudml.org/doc/73992>.

@article{Vincent1969,
abstract = {Let $\omega $ be an open relatively compact weakly determining subset of a locally compact harmonic space in the axiomatic of Boboc-Constantinescu-Cornea. If $f$ is continuous on $\overline\{\omega \}$ and harmonic in $\omega $ the $f$ may be uniformly approximated on $\overline\{\omega \}$ to within $\varepsilon $ by a function harmonic in an open set containing $\overline\{\omega \}$. The proof uses an extension of the Weierstrass-Stone theorem to geometric simplexes.},
author = {Vincent-Smith, G. F.},
journal = {Annales de l'institut Fourier},
keywords = {partial differential equations},
language = {eng},
number = {2},
pages = {339-353},
publisher = {Association des Annales de l'Institut Fourier},
title = {Uniform approximation of harmonic functions},
url = {http://eudml.org/doc/73992},
volume = {19},
year = {1969},
}

TY - JOUR
AU - Vincent-Smith, G. F.
TI - Uniform approximation of harmonic functions
JO - Annales de l'institut Fourier
PY - 1969
PB - Association des Annales de l'Institut Fourier
VL - 19
IS - 2
SP - 339
EP - 353
AB - Let $\omega $ be an open relatively compact weakly determining subset of a locally compact harmonic space in the axiomatic of Boboc-Constantinescu-Cornea. If $f$ is continuous on $\overline{\omega }$ and harmonic in $\omega $ the $f$ may be uniformly approximated on $\overline{\omega }$ to within $\varepsilon $ by a function harmonic in an open set containing $\overline{\omega }$. The proof uses an extension of the Weierstrass-Stone theorem to geometric simplexes.
LA - eng
KW - partial differential equations
UR - http://eudml.org/doc/73992
ER -

References

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  1. [1] H. BAUER, Frontière de Šilov et problème de Dirichlet, Sem. Brelot Choquet Deny, 3e année, (1958-1959). 
  2. [2] H. BAUER, Minimalstellen von Functionen und Extremalpunkt II, Archiv der Math. 11, (1960), 200-203. Zbl0098.08003MR24 #A251
  3. [3] H. BAUER, Axiomatische Behandlung des Dirichletschen Problem fur elliptische und parabolische Differentialgleichungen, Math. Ann., 146 (1962) 1-59. Zbl0107.08003MR26 #1612
  4. [4] N. BOBOC, C. CONSTANTINESCU and A. CORNEA, Axiomatic theory of harmonic functions. Non negative superharmonic functions, Ann. Inst. Fourier, Grenoble, 15 (1965) 283-312. Zbl0139.06604MR32 #2603
  5. [5] N. BOBOC, and A. CORNEA, Convex cones of lower semicontinuous functions, Rev. Roum. Math. Pures et Appl. 13 (1967) 471-525. Zbl0155.17301MR35 #7113
  6. [6] M. BRELOT, Sur l'approximation et la convergence dans la théorie des fonctions harmoniques ou holomorphes, Bull. Soc. Math. France, 73 (1945) 55-70. Zbl0061.22804MR7,205a
  7. [7] M. BRELOT, Éléments de la théorie classique du potential, 2e éd. (1961) Centre de documentation universitaire, Paris. 
  8. [8] M. BRELOT, Axiomatique des fonctions harmoniques, Séminaire de mathématiques supérieures, Montréal (1965). 
  9. [9] J. DENY, Sur l'approximation des fonctions harmoniques, Bull. Soc. Math. France, 73 (1945) 71-73. Zbl0063.01088MR7,205b
  10. [10] J. DENY, Systèmes totaux de fonctions harmoniques, Ann. Inst. Fourier, Grenoble, 1 (1949) 103-113. 
  11. [11] D. A. EDWARDS, Minimum-stable wedges of semicontinuous functions, Math. Scand. 19 (1966) 15-26. Zbl0146.37002MR35 #5907
  12. [12] D. A. EDWARDS, On uniform approximation of affine functions on a compact convex set, Quart J. Math. Oxford (2), 20 (1969), 139-42. Zbl0177.16303MR40 #3285
  13. [13] D. A. EDWARDS and G. F. VINCENT-SMITH, A Weierstrass-Stone theorem for Choquet simplexes, Ann. Inst. Fourier, Grenoble, 18 (1968) 261-282. Zbl0172.15604MR39 #6060
  14. [14] R. M. HERVÉ, Développements sur une théorie axiomatique des fonctions surharmoniques, C.R. Acad. Sci. Paris, 248 (1959) 179-181. Zbl0096.30401MR21 #5097
  15. [15] R. R. PHELPS, Lectures on Choquet's theorem, van Nostrand, Princeton N. J. (1966). Zbl0135.36203MR33 #1690
  16. [16] A. de la PRADELLE, Approximation et caractère de quasi-analyticité dans la théorie axiomatique des fonctions harmoniques, Ann. Inst. Fourier, Grenoble, 17 (1967) 383-399. Zbl0153.15501MR37 #3040

Citations in EuDML Documents

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  1. Arnaud de La Pradelle, Approximation des fonctions harmoniques à l'aide d'un théorème de G. F. Vincent-Smith
  2. Gilles Tissier, Quasi-analyticité et approximation sur la frontière d'un ouvert quelconque, dans la théorie axiomatique des fonctions harmoniques
  3. Denis Feyel, A. de La Pradelle, Principe du minimum et préfaisceaux maximaux
  4. A. de La Pradelle, À propos du mémoire de Vincent-Smith sur l'approximation des fonctions harmoniques
  5. Ian Reay, Subduals and tensor products of spaces of harmonic functions

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