Previous |  Up |  Next


tempered distribution; convolution operator; Fourier transform; convergence of sequences
\font\psaci=rsfs10 \font\ppsaci=rsfs7 In this paper we show that if $S$ is a convolution operator in $\text{\ppsaci S}^{\,\, \prime }$, and $S\ast \text{\ppsaci S}^{\,\, \prime }=\text{\ppsaci S}^{\,\, \prime }$, then the zeros of the Fourier transform of $S$ are of bounded order. Then we discuss relations between the topologies of the space $\text{\psaci O}_c^{\, \prime }$ of convolution operators on $\text{\ppsaci S}^{\,\, \prime }$. Finally, we give sufficient conditions for convergence in the space of convolution operators in $\text{\ppsaci S}^{\,\, \prime }$ and in its dual.
[1] Barros-Neto J.: An Introduction to the Theory of Distributions. Marcel Dekker, New York, 1973. MR 0461128 | Zbl 0512.46040
[2] Grothendieck A.: Produits Tensoriels Topologiques et Espaces Nucleaires. Memoirs of the Amer. Math. Soc. 16, Providence, 1966. MR 1609222 | Zbl 0123.30301
[3] Hörmander L.: On the division of distributions by polynomials. Ark. Mat., Band 3, No. 53 (1958), 555-568. MR 0124734
[4] Horvath J.: Topological Vector Spaces and Distributions. Vol. I, Addison-Wesley, Mass., 1966. MR 0205028 | Zbl 0143.15101
[5] Keller K.: Some convergence properties of distributions. Studia Mathematica 77 (1983), 87-93. MR 0738046
[6] Schwartz L.: Théorie des Distributions. Hermann, Paris, 1966. MR 0209834 | Zbl 0962.46025
[7] Sznajder S., Zielezny Z.: On some properties of convolution operators in $\mathscr{K}_{1}^{\prime}$ and $\mathscr{S}^{\prime}$. J. Math. Anal. Appl. 65 (1978), 543-554. MR 0510469
Partner of
EuDML logo