Acta Mathematica Sinica, English Series

, Volume 34, Issue 6, pp 1001–1014 | Cite as

The Lp,q-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces Lp,q(ℝd+1)

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Abstract

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, \(\sum\nolimits_{k \in \mathbb{Z}^{d + 1} } {\left| {\hat \varphi (\xi + 2\pi k)} \right|^2 > 0}\). Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results.

Keywords

Mixed Lebesgue spaces Lp,q-stability semi-convolution 

MR(2010) Subject Classification

46B15 42C15 42C40 41A58 

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Notes

Acknowledgements

We thank the referees very much for elaborate and valuable suggestions which helped to improve this paper.

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceTianjin University of TechnologyTianjinP. R. China
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinP. R. China

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