Abstract
<p style='text-indent:20px;'>We give four generalizations of the classical Lebesgue's theorem to multi-dimensional functions and Fourier series. We introduce four new concepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four different types of summability and convergences investigated in the literature, we prove that the Cesàro means <inline-formula><tex-math id="M1">\begin{document}$ \sigma_n^{\alpha}f $\end{document}</tex-math></inline-formula> of the Fourier series of a multi-dimensional function converge to <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> at each Lebesgue point as <inline-formula><tex-math id="M3">\begin{document}$ n\to \infty $\end{document}</tex-math></inline-formula>.</p>
Publisher
American Institute of Mathematical Sciences (AIMS)
Reference60 articles.
1. J. Arias de Reyna.Pointwise convergence of fourier series, J. London Math. Soc., 65 (2002), 139-153.
2. N. K. Bary, A Treatise on Trigonometric Series, Vols. I, II. Authorized translation by Margaret F. Mullins. A Pergamon Press Book The Macmillan Company, New York 1964.
3. E. S. Belinsky, Summability of multiple Fourier series at Lebesgue points, Teor. Funkci$\mathop l\limits^ \vee $ Funkcional. Anal. i Priložen, 169 (1975), 3–12, (Russian).
4. H. Berens, Z. Li, Y. Xu.On $l_1$ Riesz summability of the inverse Fourier integral, Indag. Math. (N.S.), 12 (2001), 41-53.
5. H. Berens, Y. Xu.Fejér means for multivariate Fourier series, Math. Z., 221 (1996), 449-465.
Cited by
4 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献