Journal of Fourier Analysis and Applications

, Volume 13, Issue 5, pp 551–576

# Distributional Point Values and Convergence of Fourier Series and Integrals

Article

DOI: 10.1007/s00041-006-6015-z

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.