On the local integrability condition for generalised translation-invariant systems

  • Jordy Timo van VelthovenEmail author


This paper considers the local integrability condition for generalised translation-invariant systems and its relation to the Calderón integrability condition, the temperateness condition and the uniform counting estimate. It is shown that sufficient and necessary conditions for satisfying the local integrability condition are closely related to lower and upper bounds on the number of lattice points that intersect with the translates of a compact set. The results are complemented by examples that illustrate the crucial interplay between the translation subgroups and the generating functions of the system.


Calderón integrability condition Frames Generalised translation-invariant systems Local integrability condition Uniform counting estimate 

Mathematics Subject Classification

42C40 43A32 42C15 43A70 



The author thanks José Luis Romero for useful discussions and for his help with several of the examples. Thanks also goes to Peter Kuleff and Jakob Lemvig for reading the manuscript and providing helpful comments.


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© Universitat de Barcelona 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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