On convergence almost everywhere of series of dilated functions

Michel J.G. Weber

doi:10.1016/j.crma.2015.07.010

Number theory/Harmonic analysis

On convergence almost everywhere of series of dilated functions

Jean-Pierre Kahane

Michel J.G. Weber ¹

¹IRMA, 10, rue du Général-Zimmer, 67084 Strasbourg cedex, France

Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 883-886

Abstract (Original language)
Résumé

Let $f (x) = \sum_{ℓ \in Z} a_{ℓ} e^{2 i π ℓ x}$ , where $\sum_{k \geq 1} a_{k}^{2} d (k) < \infty$ and $d (k) = \sum_{d | k} 1$ and let $f_{n} (x) = f (n x)$ . We show by using a new decomposition of squared sums that, for any $K \subset N$ finite, ${‖ \sum_{k \in K} c_{k} f_{k} ‖}_{2}^{2} \leq (\sum_{m = 1}^{\infty} a_{m}^{2} d (m)) \sum_{k \in K} c_{k}^{2} d (k^{2})$ . If $f^{s} (x) = \sum_{j = 1}^{\infty} \frac{\sin 2 π j x}{j^{s}}$ , $s > 1 / 2$ , by only using elementary Dirichlet convolution calculus, we show that for $0 < ε \leq 2 s - 1$ , $ζ {(2 s)}^{- 1} {‖ \sum_{k \in K} c_{k} f_{k}^{s} ‖}_{2}^{2} \leq \frac{1 + ε}{ε} (\sum_{k \in K} {| c_{k} |}^{2} σ_{1 + ε - 2 s} (k))$ , where $σ_{h} (n) = \sum_{d | n} d^{h}$ . From this, we deduce that if $f \in BV (T)$ , $〈 f, 1 〉 = 0$ and $\sum_{k = 1}^{\infty} c_{k}^{2} \frac{{(\log \log k)}^{4}}{{(\log \log \log k)}^{2}} < \infty$ , then the series $\sum_{k} c_{k} f_{k}$ converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc ([1], th. 3) ( $n_{k} = k$ ), where it was assumed that $\sum_{k = 1}^{\infty} c_{k}^{2} {(\log \log k)}^{γ}$ converges for some $γ > 4$ .

Soit $f (x) = \sum_{ℓ \in Z} a_{ℓ} e^{2 i π ℓ x}$ telle que la série $\sum_{k \geq 1} a_{k}^{2} d (k)$ où $d (k) = \sum_{d | k} 1$ converge, et soit $f_{n} (x) = f (n x)$ . Nous montrons à l'aide d'une nouvelle décomposition des sommes carrées que ${‖ \sum_{k \in K} c_{k} f_{k} ‖}_{2}^{2} \leq (\sum_{m = 1}^{\infty} a_{m}^{2} d (m)) \sum_{k \in K} c_{k}^{2} d (k^{2})$ , pour tout ensemble fini d'entiers K. Si $f^{s} (x) = \sum_{j = 1}^{\infty} \frac{\sin 2 π j x}{j^{s}}$ , $s > 1 / 2$ , nous montrons aussi, par un calcul simple sur les convolutions de Dirichlet, que $ζ {(2 s)}^{- 1} {‖ \sum_{k \in K} c_{k} f_{k}^{s} ‖}_{2}^{2} \leq \frac{1 + ε}{ε} (\sum_{k \in K} {| c_{k} |}^{2} σ_{1 + ε - 2 s} (k))$ , où $0 < ε \leq 2 s - 1$ et $σ_{h} (n) = \sum_{d | n} d^{h}$ . Nous en déduisons que, pour tout $f \in BV (T)$ telle que $〈 f, 1 〉 = 0$ , si la série $\sum_{k = 1}^{\infty} c_{k}^{2} \frac{{(\log \log k)}^{4}}{{(\log \log \log k)}^{2}}$ converge, alors la série $\sum_{k} c_{k} f_{k}$ converge presque partout. Cela améliore un résultat récent, dépendant d'une analyse fine sur le polydisque ([1], th. 3) ( $n_{k} = k$ ), où l'on suppose que la série $\sum_{k = 1}^{\infty} c_{k}^{2} {(\log \log k)}^{γ}$ converge pour un réel $γ > 4$ .

Article information
Cite this article

Received: 2014-07-28
Accepted: 2015-07-27
Published online: 2015-08-29

DOI: 10.1016/j.crma.2015.07.010

Author's affiliations:

Michel J.G. Weber ¹

¹ IRMA, 10, rue du Général-Zimmer, 67084 Strasbourg cedex, France

Michel J.G. Weber. On convergence almost everywhere of series of dilated functions. Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 883-886. doi: 10.1016/j.crma.2015.07.010

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References
Cited by

[1] C. Aistleitner; I. Berkes; K. Seip GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc., Volume 17 (2015) no. 6, pp. 1517-1546

[2] C. Aistleitner; I. Berkes; K. Seip; M. Weber Convergence of series of dilated functions and spectral norms of GCD matrices, Acta Arith., Volume 168 (2015) no. 3, pp. 221-246

[3] I. Berkes; M. Weber On the convergence of $\sum c_{k} f (n_{k} x)$ , Mem. Amer. Math. Soc., Volume 201 (2009) no. 943 (vi+72)

[4] I. Berkes; M. Weber On series of dilated functions, Q. J. Math., Volume 65 (2014) no. 1, p. 2552

[5] T. Hilberdink An arithmetical mapping and applications to Ω-results for the Riemann Zeta function, Acta Arith., Volume 139 (2009), pp. 341-367

[6] P. Lindqvist; K. Seip Note on some greatest common divisor matrices, Acta Arith., Volume LXXXIV (1998) no. 2, pp. 149-154

[7] M. Weber On systems of dilated functions, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011), pp. 1261-1263

[8] M. Weber An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function, 2014 | arXiv

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