ACTA issues

Pointwise convergence of series conjugate to double Fourier series

Árpád Jenei, Ferenc Móricz

Acta Sci. Math. (Szeged) 72:3-4(2006), 555-567

Abstract. We give sufficient conditions for the convergence of the symmetric as well as unsymmetric rectangular partial sums of the series conjugate to 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 series conjugate to 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 or Zygmund classes in two variables.

AMS Subject Classification (1991): 42A50, 42B05

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

Received January 26, 2006. (Registered under 5937/2009.)