On the convergence of <IMG ALIGN=MIDDLE > and the Lip 1/2 class期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

On the convergence of and the Lip 1/2 class

Authors:	Istvá n Berkes

Affiliation:	Mathematical Institute of the Hungarian Academy of Sciences, H-1364 Budapest, P.O.B. 127, Hungary

Abstract:	We investigate the almost everywhere convergence of $sum c_{n} f(nx)$ , where is a measurable function satisfying $begin{equation}f(x+1) = f(x), qquad int _{0}^{1} f(x) , dx =0.end{equation}$ By a known criterion, if satisfies the above conditions and belongs to the Lip class for some , then $sum c_{n} f(nx)$ is a.e. convergent provided $sum c_{n}^{2} < +infty$ . Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions and almost exponentially growing sequences $(n_{k})$ such that $sum c_{k} f(n_{k} x)$ is a.e. divergent for some $(c_{k})$ with $sum c_{k}^{2} < +infty$ . For functions with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.

Keywords:	Almost everywhere convergence Lipschitz classes lacunary series law of the iterated logarithm

	点击此处可从《Transactions of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Transactions of the American Mathematical Society》下载全文