The Summability Of Fourier Series On T~N And Its Applications

The Fourier series of a continuous function may diverge at some points, and there is an integrable function whose Fourier series is divergent everywhere, so it’s crucial to research the summability and application of Fourier series.This article is divided into three chapters on the content. Firstly it gives the back-ground of Cesaro and Abel summation method of Fourier series, introduces Dirichlet, Fejer kernel and Poission kernel on T, mainly gives the summability of Fourier series on T.Secondly, it gives the concept of N-torus TN, introduces trigonometric polynom-ial, trigonometric series, Dirichlet, Fejer kernel and Poission kernel on TN, mainly promotes the summability of Fourier series of a continuous function, LP(T)(1≤p<∞) function and L∞function on T to TN.Finally, it gives two sufficient conditions which can judge trigonometric series for the Fourier series, researches the necessary and sufficient conditions which can judge a trigonometric series for the Fourier series of a continuous function, LI(T)function, LP(T)(1<p<∞) function by using the conclusions in the first two chapters about Cesaro and Abel summation method of Fourier series, and promotes the conclusions to TN.