Generalized Cesàro summation of multiple Fourier series (Q1866878)

Let \(f\in L(-\pi,\pi)^m\) and the boundary set \(U(f)\subset Z^m\cap [0,\infty)^m\). The set \(U\) belongs to the class \(A_1\) if the point \(\mathbf k = (k_1,\dots,k_m)\in U \) implies that the integral rectangle \[ \Bigg(\prod\limits_{j=1}^m [0,k_j]\Bigg) \cap Z^m\subseteq U. \] The main result of the paper is as follows. There exists a constant \(B>0\), which depends only on the space dimensions, such that for any \(U\in A_1\) there exists the estimate of the norm of the corresponding operator \[ \|\sigma_{U(f)}\|_{L\to L} = \;|\sigma_{U(f)}\|_{C\to C} \leq B. \]