A new look at old theorems of Fejér and Hardy (Q6150807)

Several results on trigonometric Fourier series are presented. Firstly, the convergence of trigonometric Fourier series via a new Tauberian theorem for Cesàro summable series in abstract normed spaces is studied. Then, sufficient conditions for the convergence of trigonometric Fourier series in homogeneous Banach spaces over the circle are obtained. Among others, an extended classical Fejér's theorem on the uniform Cesàro summability of the Fourier series on sets using a refined version of Cantor's theorem on the uniform continuity of a mapping between metric spaces is established. Finally, a generalization of the classical Hardy theorem on the uniform convergence of the Fourier series on sets is proved.