Hilbert transform and singular integrals on the spaces of tempered ultradistributions

Andrzej Kamiński; Dušanka Perišić; Stevan Pilipović

Banach Center Publications (2000)

  • Volume: 53, Issue: 1, page 139-153
  • ISSN: 0137-6934

Abstract

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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.

How to cite

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Kamiński, Andrzej, Perišić, Dušanka, and Pilipović, Stevan. "Hilbert transform and singular integrals on the spaces of tempered ultradistributions." Banach Center Publications 53.1 (2000): 139-153. <http://eudml.org/doc/209069>.

@article{Kamiński2000,
abstract = {The Hilbert transform on the spaces $S^\{\prime \}*(R^d)$ 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 $S^\{\prime \}*(R^d)$, whose special cases are the Hilbert transform and Riesz operators.},
author = {Kamiński, Andrzej, Perišić, Dušanka, Pilipović, Stevan},
journal = {Banach Center Publications},
keywords = {Hilbert transform; spaces of tempered ultra-distributions; singular integral operators; Riesz operators},
language = {eng},
number = {1},
pages = {139-153},
title = {Hilbert transform and singular integrals on the spaces of tempered ultradistributions},
url = {http://eudml.org/doc/209069},
volume = {53},
year = {2000},
}

TY - JOUR
AU - Kamiński, Andrzej
AU - Perišić, Dušanka
AU - Pilipović, Stevan
TI - Hilbert transform and singular integrals on the spaces of tempered ultradistributions
JO - Banach Center Publications
PY - 2000
VL - 53
IS - 1
SP - 139
EP - 153
AB - The Hilbert transform on the spaces $S^{\prime }*(R^d)$ 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 $S^{\prime }*(R^d)$, whose special cases are the Hilbert transform and Riesz operators.
LA - eng
KW - Hilbert transform; spaces of tempered ultra-distributions; singular integral operators; Riesz operators
UR - http://eudml.org/doc/209069
ER -

References

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