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2000 | 53 | 1 | 139-153
Tytuł artykułu

Hilbert transform and singular integrals on the spaces of tempered ultradistributions

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The Hilbert transform on the spaces $S'*(R^d)$ of tempered ultradistributions is defined, uniquely in the sense of hyperfunctions, as the composition of the classical Hilbert transform with the operators of multiplying and dividing a function by a certain elliptic ultrapolynomial. We show that the Hilbert transform of tempered ultradistributions defined in this way preserves important properties of the classical Hilbert transform. We also give definitions and prove properties of singular integral operators with odd and even kernels on the spaces $S'*(R^d)$, whose special cases are the Hilbert transform and Riesz operators.
Słowa kluczowe
Rocznik
Tom
53
Numer
1
Strony
139-153
Opis fizyczny
Daty
wydano
2000
Twórcy
  • Institute of Mathematics, University of Rzeszów, Rejtana 16 C, 35-310 Rzeszów, Poland
  • Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia
  • Institute of Mathematics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Yugoslavia
Bibliografia
  • [1] E. J. Beltrami and M. R. Wohlers, Distributions and Boundary Values of Analytic Functions, Academic Press, New York, 1966.
  • [2] S. Ishikawa, Generalized Hilbert transforms in tempered distributions, Tokyo J. Math. 10 (1987), 119-132.
  • [3] A. Kamiński, D. Perišić and S. Pilipović, Integral transforms on the spaces of tempered ultradistributions, Demonstratio Math. 33 (2000), to appear.
  • [4] S. Koizumi, On the singular integrals I-VI, Proc. Japan Acad. 34 (1958), 193-198; 235-240; 594-598; 653-656; 35 (1959), 1-6; 323-328.
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  • [6] H. Komatsu, Ultradistributions, I, J. Fac. Sci. Univ. Tokyo Sect. IA 20 (1973), 25-105.
  • [7] H. Komatsu, Ultradistributions, II, J. Fac. Sci. Univ. Tokyo Sect. IA 24 (1977), 607-628.
  • [8] D. Kovačević and S. Pilipović, Structural properties of the spaces of tempered ultradistributions, in: Complex Analysis and Generalized Functions, Varna 1991, Publ. House of the Bugarian Academy of Sciences, Sofia 1993, 169-184.
  • [9] J. N. Pandey, An extension of the Gelfand-Shilov technique for Hilbert transforms, Journal of Applicable Analysis 13 (1982), 279-290.
  • [10] B. E. Petersen, Introduction to the Fourier Transform & Pseudo-differential Operators, Pitman, Boston, 1983.
  • [11] S. Pilipović, Hilbert transformation of Beurling ultradistributions, Rend. Sem. Mat. Univ. Padova 77 (1987), 1-13.
  • [12] S. Pilipović, Tempered ultradistributions, Boll. Un. Mat. Ital. (7) 2-B (1988), 235-251.
  • [13] S. Pilipović, Beurling-Gevrey tempered ultradistributions as boundary values, Portug. Math. 48 (1991), 483-504.
  • [14] S. Pilipović, Characterization of bounded sets in spaces of ultradistributions, Proc. Amer. Math. Soc. 120 (1994), 1191-1206.
  • [15] O. P. Singh and J. N. Pandey, The n-dimensional Hilbert transform of distributions, its inversion and applications, Can. J. Math 42 (1990), 239-258.
  • [16] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971.
  • [17] H.-G. Tillmann, Randverteilungen analytischer Funktionen und Distributionen, Math. Zeitschr. 59 (1953), 61-83.
  • [18] V. S. Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow, 1979.
  • [19] J. Wloka, Grundräume und verallgemeinerte Funktionen, Lecture Notes in Math. 82, Springer, Berlin, 1969.
  • [20] B. Ziemian, The modified Cauchy transformation with applications to generalized Taylor expansions, Studia Math. 102 (1992), 1-24.
Typ dokumentu
Bibliografia
Identyfikatory
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bwmeta1.element.bwnjournal-article-bcpv53z1p139bwm
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