On convergence almost everywhere of series of dilated functions
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- 作者：Michel J.G. Weber m.j.g.weber@gmail.com" class="auth_mail" title="E-mail the corresponding author
- 刊名：Comptes Rendus Mathematique
- 出版年：2015
- 出版时间：October 2015
- 年：2015
- 卷：353
- 期：10
- 页码：883-886
- 全文大小：224 K
文摘
Let f(x)=∑鈩?isin;Za鈩?/sub>e2i蟺鈩搙, where 
and d(k)=∑d|k1 and let fn(x)=f(nx). We show by using a new decomposition of squared sums that, for any K⊂N finite, . If , s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<蔚≤2s−1, , where 蟽h(n)=∑d|ndh. From this, we deduce that if f∈BV(T), 銆坒,1銆?0 and , 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 converges for some 纬>4.
and d(k)=∑d|k1 and let fn(x)=f(nx). We show by using a new decomposition of squared sums that, for any K⊂N finite, . If , s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<蔚≤2s−1, , where 蟽h(n)=∑d|ndh. From this, we deduce that if f∈BV(T), 銆坒,1銆?0 and , 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 converges for some 纬>4.