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On the Convergence of ∑ cnf(nx) and the lip 1/2 Class

Istvan Berkes
Transactions of the American Mathematical Society
Vol. 349, No. 10 (Oct., 1997), pp. 4143-4158
Stable URL: http://www.jstor.org/stable/2155578
Page Count: 16
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the Convergence of ∑ cnf(nx) and the lip 1/2 Class
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Abstract

We investigate the almost everywhere convergence of Σ cn f(nx), where f is a measurable function satisfying $f(x + 1) = f(x),\quad \int^1_0 f(x) dx = 0.$ By a known criterion, if f satisfies the above conditions and belongs to the Lip α class for some $\alpha > 1/2$, then ∑ cn f(nx) is a.e. convergent provided $\sum c^2_n < +\infty$. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions f and almost exponentially growing sequences (nk) such that ∑ ck f(nk x) is a.e. divergent for some (ck) with $\Sigma c^2_k < +\infty$. For functions f 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.

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