On some properties of multiple Fourier series and transforms (Q1813236)

Given a positive monotone decreasing function \(\varphi\) on \([0, \infty)\), a summability method for a multiple Fourier series (of a function \(f\in L(\mathbb{T}^ m)\)) is defined by \(\sigma^ \varphi_ r(x)= (1/\varphi(r)) \sum^ r_{\nu= 0} \varphi(r- \nu) S_ \nu(x)\), where \(S_ \nu(x)\) are the quadratic partial sums of the Fourier series and \(\varphi(r)= \sum^ r_{\nu= 0} \varphi(\nu)\). It is proved that under the regularity condition \(\inf_{\nu\geq 1} \varphi(2\nu)/\varphi(\nu)= \theta> 1\) the sums \(\sigma^ \varphi_ r(x)\) converge to \(f(x)\) a.e. on \(\mathbb{T}^ m\), when \(r\to \infty\). The second part of the paper deals with some properties of Fourier transforms.