The modified Cauchy transformation with applications to generalized Taylor expansions

Bogdan Ziemian

Studia Mathematica (1992)

  • Volume: 102, Issue: 1, page 1-24
  • ISSN: 0039-3223

Abstract

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We generalize to the case of several variables the classical theorems on the holomorphic extension of the Cauchy transforms. The Cauchy transformation is considered in the setting of tempered distributions and the Cauchy kernel is modified to a rapidly decreasing function. The results are applied to the study of "continuous" Taylor expansions and to singular partial differential equations.

How to cite

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Ziemian, Bogdan. "The modified Cauchy transformation with applications to generalized Taylor expansions." Studia Mathematica 102.1 (1992): 1-24. <http://eudml.org/doc/215910>.

@article{Ziemian1992,
abstract = {We generalize to the case of several variables the classical theorems on the holomorphic extension of the Cauchy transforms. The Cauchy transformation is considered in the setting of tempered distributions and the Cauchy kernel is modified to a rapidly decreasing function. The results are applied to the study of "continuous" Taylor expansions and to singular partial differential equations.},
author = {Ziemian, Bogdan},
journal = {Studia Mathematica},
keywords = {several variables; holomorphic extension of the Cauchy transforms; tempered distributions; Cauchy kernel; rapidly decreasing function; Taylor expansions; singular partial differential equations},
language = {eng},
number = {1},
pages = {1-24},
title = {The modified Cauchy transformation with applications to generalized Taylor expansions},
url = {http://eudml.org/doc/215910},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Ziemian, Bogdan
TI - The modified Cauchy transformation with applications to generalized Taylor expansions
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 1
SP - 1
EP - 24
AB - We generalize to the case of several variables the classical theorems on the holomorphic extension of the Cauchy transforms. The Cauchy transformation is considered in the setting of tempered distributions and the Cauchy kernel is modified to a rapidly decreasing function. The results are applied to the study of "continuous" Taylor expansions and to singular partial differential equations.
LA - eng
KW - several variables; holomorphic extension of the Cauchy transforms; tempered distributions; Cauchy kernel; rapidly decreasing function; Taylor expansions; singular partial differential equations
UR - http://eudml.org/doc/215910
ER -

References

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  1. [1] H. Bremermann, Distributions, Complex Variables and Fourier Transform, Addison-Wesley, 1965. 
  2. [2] A. Kaneko, Introduction to Hyperfunctions, Math. Appl., Kluwer, Dordrecht 1988. Zbl0687.46027
  3. [3] H. Komatsu, An introduction to the theory of hyperfunctions, in: Lecture Notes in Math. 287, Springer, 1973, 1-43. 
  4. [4] H. M. Reimann, Transformation de Fourier et intégrales singulières, Cours d'analyse harmonique 1982/83, Université de Berne. 
  5. [5] W. Rudin, Lectures on the Edge-of-the-Wedge Theorem, CBMS Regional Conf. Ser. in Math. 6, Amer. Math. Soc., 1971. 
  6. [6] J. Schmets, Hyperfonctions et microfonctions d'une variable, Publications d'Institut de Mathématique, Université de Liège, 1979-1980. 
  7. [7] Z. Szmydt, The Paley-Wiener theorem for the Mellin transformation, Ann. Polon. Math. 51 (1990), 313-324. Zbl0733.46020
  8. [8] Z. Szmydt and B. Ziemian, Multidimensional Mellin transformation and partial differential operators with regular singularities, Bull. Polish Acad. Sci. Math. 35 (1987), 167-180. 
  9. [9] Z. Szmydt and B. Ziemian, Solutions of singular elliptic equations via the Mellin transformation on sets of high order of tangency to the singular lines, ibid. 36 (1988), 521-535. Zbl0777.35025
  10. [10] Z. Szmydt and B. Ziemian, Local existence and regularity of solutions of singular elliptic operators on manifolds with corner singularities, J. Differential Equations 23 (1990), 1-25. Zbl0702.35105
  11. [11] Z. Szmydt and B. Ziemian, Characterization of Mellin distributions supported by certain noncompact sets, this issue, 25-38. 
  12. [12] Z. Szmydt and B. Ziemian, The Mellin Transformation and Fuchsian Type Partial Differential Equations, book to be published by Kluwer Academic Publishers. Zbl0771.35002
  13. [13] B. Ziemian, An analysis of microlocal singularities of functions and distributions on the real line, Bull. Polish Acad. Sci. Math. 32 (1984), 157-164. Zbl0584.46027
  14. [14] B. Ziemian, Taylor formula for distributions in several dimensions, ibid. 34 (1986), 277-286. Zbl0632.46033
  15. [15] B. Ziemian, Taylor formula for distributions, Dissertationes Math. 264 (1988). 
  16. [16] B. Ziemian, The Mellin transformation and multidimensional generalized Taylor expansions of singular functions, J. Fac. Sci. Univ. Tokyo 36 (1989), 263-295. Zbl0713.46025
  17. [17] B. Ziemian, Elliptic corner operators in spaces with continuous radial asymptotics I, J. Differential Equations, to appear. Zbl0777.47028
  18. [18] B. Ziemian, Elliptic corner operators in spaces with continuous radial asymptotics II, in: Banach Center Publ. 27, to appear. Zbl0813.47061
  19. [19] B. Ziemian, Continuous radial asymptotic for solutions to elliptic Fuchsian equations in 2 dimensions, in: Proc. Sympos. Microlocal Analysis and its Applications, RIMS Kokyuroku 750, Kyoto Univ., 1991, 3-19. 

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