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Diophantine approximations and the sets of divergence of some Fourier series. (English. Russian original) Zbl 0863.42007
Russ. Acad. Sci., Dokl., Math. 49, No. 3, 471-473 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 336, No. 2, 151-153 (1994).
Let \(f(x)\) be a continuous \(2\pi\)-periodic function whose Fourier coefficients satisfy the condition \[ \sum^\infty_{n=1}(|a_n|+|b_n|)<\infty.\tag{1} \] Let \(M\) denote the set of \(\theta\in\mathbb{R}\) for which the series \(\sum^\infty_{n=1}{(-1)^n\over n} f(n\theta)\) diverges and \(h_\alpha(M)\) is the Hausdorff \(\alpha\)-measure of \(M\).
Theorem 1. Let the Fourier coefficients of the function \(f(x)\) be monotone decreasing and satisfy (1). Then \(h_\alpha(M)=0\) for any \(\alpha>0\).
Theorem 2. There exist functions \(f(x)\) with (1) such that \(h_\alpha(M)>0\) for any \(0<\alpha<\alpha_0(f)\).
The condition (1) is less stringent than the condition by Rao.
MSC:
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A20 Convergence and absolute convergence of Fourier and trigonometric series
11J71 Distribution modulo one
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