# Distributional Point Values and Convergence of Fourier Series and Integrals

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## Abstract

In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If \(f\in\mathcal{S}^{\prime}\left(\mathbb{R}\right)\) and \(x_{0}\in\mathbb{R}\), and \(\widehat{f}\) is locally integrable, then \(f(x_{0})=\gamma\ \) distributionally if and only if there exists k such that \(\frac{1}{2\pi}\lim_{x\rightarrow\infty}\int_{-x}^{ax} \hat{f}(t)e^{-ix_{0}t}\,\mathrm{d}t=\gamma\ \ (\mathrm{C},k)\,\), for each a > 0, and similarly in the case when \(\widehat{f}\) is a general distribution. Here \((\mathrm{C},k)\) means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by \(\lim_{x\rightarrow\infty}\sum_{-x\leq n\leq ax}a_{n}e^{inx_{0}}=\gamma\ (\mathrm{C},k)\,\). We also show that under some extra conditions, as if the sequence \(\left\{a_{n}\right\} _{n=-\infty}^{\infty}\) belongs to the space \(l^{p}\) for some \(p\in\lbrack1,\infty)\) and the tails satisfy the estimate \(\sum_{\left\vert n\right\vert \geq N}^{\infty}\left\vert a_{n}\right\vert ^{p}=O\left(N^{1-p}\right) \),\ as \(N\rightarrow\infty\), the asymmetric partial sums\ converge to \(\gamma\). We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.

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