ACTA issues

Extensions of the theorems of Szász and Zygmund on the absolute convergence of Fourier series

Antal Veres

Acta Sci. Math. (Szeged) 74:1-2(2008), 191-206

Abstract. We consider the double Fourier series of functions $f\colon{\msbm T}^2\rightarrow{\msbm C}$, where ${\msbm T}^2$ is the two-dimensional torus. We prove sufficient conditions on the convergence of the double series whose terms are the $\beta $-th power of the absolute value of the Fourier coefficients of the function $f$ in question. These conditions are given in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivate in case $f$ is an absolutely continuous function. Our results extend the classical theorems of O. Szász and A. Zygmund from single to double Fourier series.

AMS Subject Classification (1991): 42A20, 42B99

Keyword(s): double Fourier series, absolute convergence, multiplicative moduli of continuity, multiplicative Lipschitz class, functions of bounded variation in the sense of Vitali and of Hardy and Krause, absolutely continuous functions of two variables, double versions of the theorems of O. Szász and A. Zygmund

Received December 11, 2006. (Registered under 6011/2009.)