On The Problems Of Almost Everywhere Convergence Of Fourier Series

The theory of Fourier series is mainly about convergent problem. Swedish mathe-matician L.Carleson proved that Fourier series of L2 functions on [0,2π] are convergen-t almost everywhere on [0,2π], which is the guess proposed by Russian mathematician Luzin. After that, it was further proved by U.S. mathematician R.Hunt that Fourier series of functions in IP (1< p< ∞) space on [0,2π] are convergent almost everywhere on [0,2π]. On the other hand, Argentine mathematician C alder on proved that Fourier series of L2 functions on [0,2π] are convergent almost everywhere on [0,2π], which is equivalent to the proposition that the maximal operator of its partial sum is satisfied with (2,2) weak inequality. In the first part of this paper, we will prove that Fourier series of functions in LP (1< p<∞) space on [0,2π] are convergent almost everywhere on [0,2π]. That is to say, the maximal operator of its partial sum is satisfied with (p, p) weak inequality. In the second part, we will study the problem of divergent set, and make up the deficiency of Y.Katznelson’s book named An Introduction to Harmonic Analysis.