On Classical Studies for Summability and Convergence of Double Fourier Series

G. H. Hardy laid the foundation of classical studies on double Fourier series at the beginning of the 20th century. In this paper we are concerned not only with Fourier series but more generally with trigonometric series. We consider Norlund means and Cesaro summation method for double Fourier Series. In section 2, we investigate the classical results on the summability and the convergence of double Fourier series from G. H. Hardy to P. Sjolin in the mid-20th century. This study concerns with the $L^1(T^2)$ -convergence of double Fourier series fundamentally. In conclusion, there are the features of the classical results by comparing and reinterpreting the theorems about double Fourier series mutually.