Convergence properties of one and multidimensional trigonometric Fourier series

Abstract

In this work, first of all, we study the convergence properties of one and multidimensional trigonometric Fourier series. It begins by discussing fundamental concepts such as vec- tor spaces, function spaces, and interpolation theorems. Inspired by Fourier analysis, it explores the feasibility of extending these concepts to infinite-dimensional spaces and or- thonormal systems. The study extensively covers the convergence behavior of Fourier series in the one-dimensional case, including partial sums, Dirichlet kernels, Fourier coefficients, and convergence criteria. It also examines the concept of Fej´er means. In the multidi- mensional context, the thesis explores topics such as partial sums, norm convergence, and summability means. Throughout the thesis, we use and refer to books [1] and [2], as well as the lectures given by my supervisor in his ”Fourier series” lectures [3]. What results are not directly referred to are taken from lectures [3, 4]. No new statements or results have been formulated in the thesis.