Number theory/Harmonic analysis
On convergence almost everywhere of series of dilated functions
Comptes Rendus. Mathématique, Volume 353 (2015) no. 10, pp. 883-886.

Let f(x)=Zae2iπx, where k1ak2d(k)< and d(k)=d|k1 and let fn(x)=f(nx). We show by using a new decomposition of squared sums that, for any KN finite, kKckfk22(m=1am2d(m))kKck2d(k2). If fs(x)=j=1sin2πjxjs, s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<ε2s1, ζ(2s)1kKckfks221+εε(kK|ck|2σ1+ε2s(k)), where σh(n)=d|ndh. From this, we deduce that if fBV(T), f,1=0 and k=1ck2(loglogk)4(logloglogk)2<, then the series kckfk converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc ([1], th. 3) (nk=k), where it was assumed that k=1ck2(loglogk)γ converges for some γ>4.

Soit f(x)=Zae2iπx telle que la série k1ak2d(k)d(k)=d|k1 converge, et soit fn(x)=f(nx). Nous montrons à l'aide d'une nouvelle décomposition des sommes carrées que kKckfk22(m=1am2d(m))kKck2d(k2), pour tout ensemble fini d'entiers K. Si fs(x)=j=1sin2πjxjs, s>1/2, nous montrons aussi, par un calcul simple sur les convolutions de Dirichlet, que ζ(2s)1kKckfks221+εε(kK|ck|2σ1+ε2s(k)), où 0<ε2s1 et σh(n)=d|ndh. Nous en déduisons que, pour tout fBV(T) telle que f,1=0, si la série k=1ck2(loglogk)4(logloglogk)2 converge, alors la série kckfk converge presque partout. Cela améliore un résultat récent, dépendant d'une analyse fine sur le polydisque ([1], th. 3) (nk=k), où l'on suppose que la série k=1ck2(loglogk)γ converge pour un réel γ>4.

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DOI: 10.1016/j.crma.2015.07.010
Weber, Michel J.G. 1

1 IRMA, 10, rue du Général-Zimmer, 67084 Strasbourg cedex, France
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Weber, Michel J.G. 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. http://www.numdam.org/articles/10.1016/j.crma.2015.07.010/

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