caa a22 4500 463247019 CHVBK 20180405153328.0 cr unu---uuuuu 170326e20071001xx s 000 0 eng 10.1007/s00041-006-6015-z doi (NATIONALLICENCE)springer-10.1007/s00041-006-6015-z Distributional Point Values and Convergence of Fourier Series and Integrals [Elektronische Daten] [Jasson Vindas, Ricardo Estrada] 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. Birkhauser Boston, 2007 Vindas Jasson Mathematics Department, Louisiana State University, Baton Rouge, Louisiana, USA aut Estrada Ricardo Mathematics Department, Louisiana State University, Baton Rouge, Louisiana, USA aut Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/5(2007-10-01), 551-576 1069-5869 13:5<551 2007 13 41 https://doi.org/10.1007/s00041-006-6015-z text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00041-006-6015-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Vindas Jasson Mathematics Department, Louisiana State University, Baton Rouge, Louisiana, USA aut NATIONALLICENCE 700 1- Estrada Ricardo Mathematics Department, Louisiana State University, Baton Rouge, Louisiana, USA aut NATIONALLICENCE 773 0- Journal of Fourier Analysis and Applications Birkhäuser-Verlag; www.birkhauser.com 13/5(2007-10-01), 551-576 1069-5869 13:5<551 2007 13 41 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer