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Characterization of the Fourier series of a distribution having a value at a point. (English) Zbl 0843.46024

Summary: Let \(f\) be a periodic distribution of period \(2\pi\). Let \(\sum^\infty_{n= -\infty} a_n e^{in\theta}\) be its Fourier series. We show that the distributional point value \(f(\theta_0)\) exists and equals \(\gamma\) if and only if the partial sums \(\Sigma_{- \infty\leq n\leq ax} a_n e^{in\theta_0}\) converge to \(\gamma\) in the Cesàro sense as \(x\to \infty\) for each \(a> 0\).

MSC:

46F10 Operations with distributions and generalized functions
42A24 Summability and absolute summability of Fourier and trigonometric series
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