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Representing Systems of Dilations and Translations in Symmetric Function Spaces

  • Sergey V. AstashkinEmail author
  • Pavel A. Terekhin
Article
  • 34 Downloads

Abstract

Let X be an arbitrary separable symmetric function space on [0, 1]. By using a combination of the frame approach and the notion of the multiplicator space \(\mathscr {M}(X)\) of X with respect to the tensor product, we investigate the problem when the sequence of dyadic dilations and translations of a function \(f\in X\) is a representing system in the space X. The main result reads that this holds whenever \(\int _0^1 f(t)\,dt\ne 0\) and \(f\in \mathscr {M}(X).\) Moreover, the condition \(f\in \mathscr {M}(X)\) turns out to be sharp in a certain sense. In particular, we prove that a decreasing nonnegative function f\(f\ne 0,\) from a Lorentz space \(\varLambda _{\varphi }\) generates an absolutely representing system of dyadic dilations and translations in \(\varLambda _{\varphi }\) if and only if \(f\in \mathscr {M}(\varLambda _{\varphi }).\)

Keywords

Sequence of dilations and translations Symmetric function space Representing system Tensor product Frame Lorentz space 

Mathematics Subject Classification

Primary 46E30 Secondary 46B70 42C15 46B15 

Notes

Acknowledgements

The work of the Sergey V. Astashkin was supported by the Ministry of Education and Science of the Russian Federation, Project 1.470.2016/1.4 and by the RFBR Grant 18-01-00414. The work of the Pavel A. Terekhln was supported by the RFBR Grant 18-01-00414. The authors are very grateful to the referee for detailed and constructive criticism that helped us improve the presentation of the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of MathematicsSamara UniversitySamaraRussia
  2. 2.Department of Mechanics and MathematicsSaratov State UniversitySaratovRussia

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