Strong summability of Riesz means (Q1110029)

Let \(\{u_ n(x)\}\) be a complete orthonormalized system of eigenfunctions of the self-adjoint extension of Laplace operator - \(\Delta\) in N-dimensional domain \(\Omega\) with discrete spectrum, and let \(\lambda_ n=\mu_ n^ 2\) be the corresponding eigenvalues numbered in increasing order. We consider the Riesz mean of order \(s\geq 0\) of the Fourier series with respect to system \(\{u_ n(x)\}\) of function f(x) of \(L_ 2(\Omega):\) \[ \sigma^{s}_{\mu}(f,x)=\sum_{\mu_ n<\mu}(1- \frac{\mu^ 2_ n}{\mu^ 2})^ sf_ nu_ n(x). \] We find the de la Vallée-Poussin sums of the Riesz means of function f(x).