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On the convergence in $ \mathcal{S}'$


Author: Stevan Pilipović
Journal: Proc. Amer. Math. Soc. 111 (1991), 949-954
MSC: Primary 46F05
DOI: https://doi.org/10.1090/S0002-9939-1991-1050022-X
MathSciNet review: 1050022
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Abstract: We prove the following assertion: Let $ {T_j},j \in \mathbb{N}$, be a sequence in $ \mathcal{S}'$ such that $ {T_j} * \phi $ converges to 0 in $ \mathcal{S}'$ as $ j \to \infty $, for any $ \phi \in \mathcal{D}$. Then $ {T_j} \to 0$ in $ \mathcal{S}'$ as $ j \to \infty $.


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  • [1] J. Dixmier and P. Malliavin, Factorisation de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. 102 (1978), 305-330. MR 517765 (80f:22005)
  • [2] A. Friedman, Generalized functions and partial differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1963. MR 0165388 (29:2672)
  • [3] K. Keller, Some convergence properties of convolutions, Studia Math. 77 (1983), 87-94. MR 738046 (85h:46059)
  • [4] H. Petzeltová and P. Vrbová, Factorization in the algebra of rapidly decreasing functions on $ {R_n}$, Comment. Math. Univ. Carolinae 19 (1978), 489-499. MR 508956 (80f:46056)
  • [5] L. A. Rubel, W. A. Squires, and B. A. Taylor, Irreducibility of certain entire functions with applications to harmonic analysis, Ann. of Math. (2) 108 (1978), 553-567. MR 512433 (80d:32003)
  • [6] L. Schwartz, Théorie des distributions, Hermann, Paris, 1966. MR 0209834 (35:730)
  • [7] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. MR 0225131 (37:726)
  • [8] V. S. Vladimirov, Generalized functions in mathematical physics, Mir, Moscow, 1979. MR 564116 (80j:46062b)
  • [9] J. Voigt, Factorization in some Fréchet algebras of differentiable functions, Studia Math. 77 (1984), 333-348. MR 741451 (85m:46051)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1050022-X
Keywords: Tempered distributions
Article copyright: © Copyright 1991 American Mathematical Society

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