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)

  • Rui Li
  • Bei Liu
  • Rui Liu
  • Qing Yue Zhang


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.


Mixed Lebesgue spaces Lp,q-stability semi-convolution 

MR(2010) Subject Classification

46B15 42C15 42C40 41A58 


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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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