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Convergence of series of dilated functions and spectral norms of GCD matrices

Volume 168 / 2015

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber Acta Arithmetica 168 (2015), 221-246 MSC: 42A16, 42A20, 42A61, 42B05, 11A05, 15A18, 26A45. DOI: 10.4064/aa168-3-2

Abstract

We establish a connection between the $L^2$ norm of sums of dilated functions whose $j$th Fourier coefficients are $\mathcal {O}(j^{-\alpha })$ for some $\alpha \in (1/2,1)$, and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in $L^2$ and for the almost everywhere convergence of series of dilated functions.

Authors

  • Christoph AistleitnerInstitute of Mathematics A
    Graz University of Technology
    Steyrergasse 30
    8010 Graz, Austria
    e-mail
  • István BerkesInstitute of Statistics
    TU Graz
    Kopernikusgasse 24/III
    8010 Graz, Austria
    e-mail
  • Kristian SeipDepartment of Mathematical Sciences
    Norwegian University of Science
    and Technology (NTNU)
    NO-7491 Trondheim, Norway
    e-mail
  • Michel WeberIRMA
    10 rue du Général Zimmer
    67084 Strasbourg Cedex, France
    e-mail

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