Summability of double Fourier series by Nörlund methods at a point (Q1197447)

Let \(f(x,y)\) be a complex-valued function, \(2\pi\)-periodic in each variable, and \(L\)-integrable over \(T^ 2:=\{(x,y)\in\mathbb{R}^ 2: x,y\in (-\pi,\pi]\}\); for \((x,y)\in T^ 2\) let \[ \varphi_{xy}(u,v):=f(x+u,y+v)+f(x-u,y+v)+f(x+u,y-v)+f(x-u,y-v)- 4f(x,y) \] and \[ \Phi_{xy}(s,t):=\int^ s_ 0\int^ f_ 0| \varphi_{xy}(u,v)| du dv\quad (s,t>0). \] The first theorem concerns Nörlund double means generated by a non-decreasing double sequence \(p_{jk}\geq 0\) \((j,k=0,1,\dots)\), \(p_{00} >0\), with \(\Delta_{11} p_{jk}\) of fixed sign and \((m+1)(n+1)p_{mn}/P_{mn}=O(1)\), where \(P_{mn}:=\sum^ m_{j=0} \sum^ n_{k=0} p_{jk}\). It is shown that under some simple order conditions on \(\Phi\) (namely \(\Phi_{xy}(s,t)=o(st)\), \(\Phi_{xy}(s,\pi)=O(s)\), \(\Phi_{xy}(\pi,t)=O(t)\), \(s,t\to 0)\) the double Fourier series of \(f\) is Nörlund summable to \(f(x,y)\) for any \((x,y)\in T^ 2\). A second theorem obtains the same conclusion when \((p_{jk})\) is a non-increasing double sequence, provided that \((p_{jk})\) satisfies some additional conditions and the order conditions on \(\Phi_{xy}\) are suitably modified. The results generalize a number of earlier theorems.