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On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups

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Abstract

This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic analysis on topological groups. These papers are known to be classical and form the basis of numerous contemporary researches into these fields.

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    R. Ya. Grabovskaya and S. G. Krein, “On a representation of the algebra of differential operators and on the related differential equations,”Dokl. Acad. Nauk SSSR,212, No. 2, 280–284 (1973).

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    G. I. Kats, “Representations of compact ring groups,”Dokl. Acad. Nauk SSSR,145, No. 5, 989–992 (1962).

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    M. G. Krein, “On a generalization of the Plancherel theorem to the case of Fourier integrals on a commutative topological group,”Dokl. Acad. Nauk SSSR,30, No. 6, 482–486 (1941).

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    M. G. Krein, “On a general method of expansion of positive definite kernels in primary products,”Dokl. Acad. Nauk SSSR,53, No. 1, 3–6 (1946).

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    M. G. Krein, “The duality principle for a bicompact group and a square block algebra,”Dokl. Acad. Nauk SSSR,69, No. 6, 725–728 (1949).

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    M. G. Krein, “Hermitian positive kernels on homogeneous spaces. I,”Ukr. Mat. Zh.,1, No. 4, 64–98 (1949); II,Ibid.,2, No. 1, 10–59 (1950).

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    B. M. Levitan, “Theorems on representations of positive definite functions for operations of generalized shifts,”Dokl. Acad. Nauk SSSR,47, No. 3, 163–165 (1945).

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    Yu. M. Berezanskii and A. A. Kalyuzhnyi,Harmonic Analysis in Hypercomplex Systems [in Russian], Naukova Dumka, Kiev (1992).

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 198–211, March, 1994.

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Vainerman, L.I. On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups. Ukr Math J 46, 204–218 (1994). https://doi.org/10.1007/BF01062235

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Keywords

  • Harmonic Analysis
  • Topological Group