Abstract. We give sufficient conditions for the convergence of the symmetric as well as unsymmetric rectangular partial sums of the double Fourier series of a complex-valued function $f\in L^1 ({\msbm T}^2)$ at a given point $(x_0, y_0) \in{\msbm T}^2$. It turns out that this convergence essentially depends on the convergence behavior of the single Fourier series of the so-called marginal functions $f(x,y_0)$, $x\in{\msbm T}$, and $f(x_0, y)$, $y\in{\msbm T}$, at $x:= x_0$ and $y:= y_0$, respectively. Our theorems apply to functions in the multiplicative Lipschitz classes as well as Zygmund classes.

AMS Subject Classification (1991): 42B05

Keyword(s): Dini test, double Fourier series, symmetric and unsymmetric rectangular partial sums, pointwise convergence, Riemann--Lebesgue lemma, multiplicative Lipschitz classes and Zygmund classes

Received August 4, 2005, and in final form January 6, 2006. (Registered under 5914/2009.)