ACTA issues

Strong Cesàro summability and statistical limit of double Fourier integrals

Ferenc Móricz

Acta Sci. Math. (Szeged) 71:1-2(2005), 159-174

Abstract. We extend the concept and basic results on statistical limit of measurable functions of one variable to those of two variables. We discuss the close connection between strong Cesàro summability and the existence of statistical limit. As an application, we prove that if $f\in L^1\cap L_{\rm loc}^\infty({\msbm R}^2)$, then the Dirichlet integral $s_\nu(f, x_1, x_2)$ of $f$ has statistical limit as $\nu\to \infty $ at every Lebesgue point $(x_1, x_2)$ of $f$ of order 2, that is, at almost every point of ${\msbm R}^2$. This means that a function $f$ in $L^1\cap L^\infty_{\rm loc}({\msbm R}^2)$ can be reconstructed by means of its Fourier transform in terms of statistical limit.

AMS Subject Classification (1991): 40C10, 42B10

Received June 29, 2004, and in final form February 15, 2005. (Registered under 5863/2009.)