Let , where and and let . We show by using a new decomposition of squared sums that, for any finite, . If , , by only using elementary Dirichlet convolution calculus, we show that for , , where . From this, we deduce that if , and , then the series converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc ([1], th. 3) (), where it was assumed that converges for some .
Soit telle que la série où converge, et soit . Nous montrons à l'aide d'une nouvelle décomposition des sommes carrées que , pour tout ensemble fini d'entiers K. Si , , nous montrons aussi, par un calcul simple sur les convolutions de Dirichlet, que , où et . Nous en déduisons que, pour tout telle que , si la série converge, alors la série converge presque partout. Cela améliore un résultat récent, dépendant d'une analyse fine sur le polydisque ([1], th. 3) (), où l'on suppose que la série converge pour un réel .
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Michel J.G. Weber 1
@article{CRMATH_2015__353_10_883_0, author = {Michel J.G. Weber}, title = {On convergence almost everywhere of series of dilated functions}, journal = {Comptes Rendus. Math\'ematique}, pages = {883--886}, publisher = {Elsevier}, volume = {353}, number = {10}, year = {2015}, doi = {10.1016/j.crma.2015.07.010}, language = {en}, }
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.010/
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