Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 4, page 665-678
  • ISSN: 0010-2628

Abstract

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We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

How to cite

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Štěpničková, Libuše. "Pointwise and locally uniform convergence of holomorphic and harmonic functions." Commentationes Mathematicae Universitatis Carolinae 40.4 (1999): 665-678. <http://eudml.org/doc/22446>.

@article{Štěpničková1999,
abstract = {We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.},
author = {Štěpničková, Libuše},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Osgood's theorem; approximation; maximum principle; harmonic space; elliptic PDE's; Osgood's theorem; approximation; maximum principle; harmonic space; second order elliptic equations},
language = {eng},
number = {4},
pages = {665-678},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Pointwise and locally uniform convergence of holomorphic and harmonic functions},
url = {http://eudml.org/doc/22446},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Štěpničková, Libuše
TI - Pointwise and locally uniform convergence of holomorphic and harmonic functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 4
SP - 665
EP - 678
AB - We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.
LA - eng
KW - Osgood's theorem; approximation; maximum principle; harmonic space; elliptic PDE's; Osgood's theorem; approximation; maximum principle; harmonic space; second order elliptic equations
UR - http://eudml.org/doc/22446
ER -

References

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  1. Boboc N., Mustaţă P., Espaces harmoniques associés aux opérateurs différentiels linéaires du second ordre de type elliptique, Springer Berlin (1968). (1968) MR0241681
  2. Constantinescu C., Cornea A., Potential Theory on Harmonic Spaces, Springer Berlin (1972). (1972) Zbl0248.31011MR0419799
  3. Conway J.B., Functions of One Complex Variable I, Springer New York (1995). (1995) MR1344449
  4. Gardiner S.J., Goldstein M., GowriSankaran K., Global approximation in harmonic spaces, Proc. Amer. Math. Soc. 122 (1994), 213-221. (1994) Zbl0809.31006MR1203986
  5. Hervé R.M., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier 12 (1962), 415-571. (1962) MR0139756
  6. Hartogs F., Rosenthal A., Über Folgen analytischer Funktionen, Math. Ann. 100 (1928), 212-263. (1928) MR1512483
  7. Osgood W.F., Note on the functions defined by infinite series whose terms are analytic functions, Ann. of Math. 3 (1901), 25-34. (1901) MR1502274
  8. Pradelle A., Approximation et caractère de quasi-analyticit dans la thorie axiomatique des fonctions harmoniques, Ann. Inst. Fourier 17 (1967), 383-399. (1967) MR0227456
  9. Titchmarsh E.C., The Theory of Functions, Oxford University Press New York (1960). (1960) 
  10. Ullrich P., Punktweise und lokal gleichmäßige Konvergenz von Folgen holomorpher Funktionen I., Math. Semesterber. 41 (1994), 69-75. (1994) Zbl0815.30002MR1332472
  11. Ullrich P., Punktweise und lokal gleichmäßige Konvergenz von Folgen holomorpher Funktionen II., Math. Semesterber. 41 (1994), 81-87. (1994) Zbl0815.30003MR1332472

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