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Acta Mathematica Hungarica

, Volume 157, Issue 1, pp 269–279 | Cite as

On the convergence of permutations of multiple orthogonal series

  • J. OlejnikEmail author
Article
  • 31 Downloads

Abstract

We provide a characterization of coefficients of a general orthogonal double-index series which allows existence of an a.e. convergent rearrangement. Namely, we show that for any sequence of numbers \({(a_{\mathbf{n}})_{\mathbf{n}\in\mathbb{N}^d}}\) the condition \({\sum_{\mathbf{n}\in\mathbb{N}^d} a_{\mathbf{n}}^{2}{\rm log}^{2} a_{\mathbf{n}}^{2} < \infty}\) is equivalent to the existence of an injective map \({\varrho\colon\mathbb{N}^{d} \rightarrow \mathbb{N}^{d}}\) such that the multiple series \({\sum_{\mathbf{n}\in\mathbb{N}^d} a_{\varrho(\mathbf{n})}\Phi_{\varrho(\mathbf{n})}}\) converges a.e. for any orthonormal system of functions \({(\Phi_{\mathbf{n}})_{\mathbf{n}\in\mathbb{N}^d}}\). To this end, a Menshov-type lemma for arbitrary subsets of \({\mathbb{N}^d}\) is proved.

Key words and phrases

multiple orthogonal series orthogonal field a.e. convergence permutation of indices 

Mathematics Subject Classification

primary 60G60 60G17 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of ŁódźŁódźPoland

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