Geodesic
Convergence of series of dilated functions and spectral norms of GCD matrices
Christoph Aistleitner1 ; István Berkes2 ; Kristian Seip3 ; Michel Weber4
1 Institute of Mathematics A Graz University of Technology Steyrergasse 30 8010 Graz, Austria
2 Institute of Statistics TU Graz Kopernikusgasse 24/III 8010 Graz, Austria
3 Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) NO-7491 Trondheim, Norway
4 IRMA 10 rue du Général Zimmer 67084 Strasbourg Cedex, France
Acta Arithmetica, Tome 168 (2015) no. 3, pp. 221-246.

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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.
DOI : 10.4064/aa168-3-2
Keywords: establish connection between norm sums dilated functions whose jth fourier coefficients mathcal alpha alpha spectral norms certain greatest common divisor gcd matrices utilizing recent bounds these spectral norms obtain sharp conditions convergence almost everywhere convergence series dilated functions

Christoph Aistleitner 1 ; István Berkes 2 ; Kristian Seip 3 ; Michel Weber 4

1 Institute of Mathematics A Graz University of Technology Steyrergasse 30 8010 Graz, Austria
2 Institute of Statistics TU Graz Kopernikusgasse 24/III 8010 Graz, Austria
3 Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) NO-7491 Trondheim, Norway
4 IRMA 10 rue du Général Zimmer 67084 Strasbourg Cedex, France 
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Christoph Aistleitner; István Berkes; Kristian Seip; Michel Weber. Convergence of series of dilated functions and spectral norms of GCD matrices. Acta Arithmetica, Tome 168 (2015) no. 3, pp. 221-246. doi : 10.4064/aa168-3-2. http://geodesic.mathdoc.fr/articles/10.4064/aa168-3-2/

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