On tempered convolution operators

Saleh Abdullah

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 1, page 1-7
  • ISSN: 0010-2628

Abstract

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In this paper we show that if S is a convolution operator in S ' , and S * S ' = S ' , then the zeros of the Fourier transform of S are of bounded order. Then we discuss relations between the topologies of the space O c ' of convolution operators on S ' . Finally, we give sufficient conditions for convergence in the space of convolution operators in S ' and in its dual.

How to cite

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Abdullah, Saleh. "On tempered convolution operators." Commentationes Mathematicae Universitatis Carolinae 35.1 (1994): 1-7. <http://eudml.org/doc/247570>.

@article{Abdullah1994,
abstract = {In this paper we show that if $S$ is a convolution operator in $\text\{S\}^\{\,\, \prime \}$, and $S\ast \text\{S\}^\{\,\, \prime \}=\text\{S\}^\{\,\, \prime \}$, then the zeros of the Fourier transform of $S$ are of bounded order. Then we discuss relations between the topologies of the space $\text\{O\}_c^\{\, \prime \}$ of convolution operators on $\text\{S\}^\{\,\, \prime \}$. Finally, we give sufficient conditions for convergence in the space of convolution operators in $\text\{S\}^\{\,\, \prime \}$ and in its dual.},
author = {Abdullah, Saleh},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {tempered distribution; convolution operator; Fourier transform; convergence of sequences; space of convolution operators; tempered distribution; convergence of sequences; convolution operator; zeros of the Fourier transform},
language = {eng},
number = {1},
pages = {1-7},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On tempered convolution operators},
url = {http://eudml.org/doc/247570},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Abdullah, Saleh
TI - On tempered convolution operators
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 1
SP - 1
EP - 7
AB - In this paper we show that if $S$ is a convolution operator in $\text{S}^{\,\, \prime }$, and $S\ast \text{S}^{\,\, \prime }=\text{S}^{\,\, \prime }$, then the zeros of the Fourier transform of $S$ are of bounded order. Then we discuss relations between the topologies of the space $\text{O}_c^{\, \prime }$ of convolution operators on $\text{S}^{\,\, \prime }$. Finally, we give sufficient conditions for convergence in the space of convolution operators in $\text{S}^{\,\, \prime }$ and in its dual.
LA - eng
KW - tempered distribution; convolution operator; Fourier transform; convergence of sequences; space of convolution operators; tempered distribution; convergence of sequences; convolution operator; zeros of the Fourier transform
UR - http://eudml.org/doc/247570
ER -

References

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  1. Barros-Neto J., An Introduction to the Theory of Distributions, Marcel Dekker, New York, 1973. Zbl0512.46040MR0461128
  2. Grothendieck A., Produits Tensoriels Topologiques et Espaces Nucleaires, Memoirs of the Amer. Math. Soc. 16, Providence, 1966. Zbl0123.30301MR1609222
  3. Hörmander L., On the division of distributions by polynomials, Ark. Mat., Band 3, No. 53 (1958), 555-568. (1958) MR0124734
  4. Horvath J., Topological Vector Spaces and Distributions, Vol. I, Addison-Wesley, Mass., 1966. Zbl0143.15101MR0205028
  5. Keller K., Some convergence properties of distributions, Studia Mathematica 77 (1983), 87-93. (1983) MR0738046
  6. Schwartz L., Théorie des Distributions, Hermann, Paris, 1966. Zbl0962.46025MR0209834
  7. Sznajder S., Zielezny Z., On some properties of convolution operators in 𝒦 1 ' and 𝒮 ' , J. Math. Anal. Appl. 65 (1978), 543-554. (1978) MR0510469

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