Principal Submodules in the Schwartz Module


In this paper we consider the Schwartz module of entire functions of exponential type having polynomial growth along the real axis. This module is equipped with the non-metrizable locally convex topology. We establish that any principal submodule is the set of limits of converging sequences which members are polynomials multiplied by the generator of the submodule. We also obtain a weight weak localizability criterion for principal submodules and some results concerning the notion «synthesizable sequence» recently introduced by A. Baranov and Yu. Belov.

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The research is supported by the Russian Science Foundation (project no. 18-11-00002).

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Correspondence to N. F. Abuzyarova.

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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, No. 5, pp. 83–88.

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Abuzyarova, N.F. Principal Submodules in the Schwartz Module. Russ Math. 64, 74–78 (2020).

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Key words

  • entire functions
  • local description of ideals and submodules
  • Schwartz space
  • spectral synthesis