On the smoothness of mean values of functions with summable spectral expansion (Q444012)

For an arbitrary bounded domain \(\Omega\subset \mathbb R^n\) the author considers a positive self-adjoint extension of the Laplace operator \(L=-\Delta\) with discrete spectrum consisting of eigenvalues \(\lambda_k\to +\infty\) that corresponds to a complete orthonormal system of eigenfunctions \(\{u_k(x)\}\) in \(L_2(\Omega)\). In the paper is studied the behavior of an arbitrary function \(f\in L_2(\Omega)\) in a neighborhood of a point at which its spectral expansion \[ E_{\lambda}f(x)=\sum_{\lambda_k<\lambda}(f,u_k)u_k(x) \] or Riesz means of order \(s\geq 0\) \[ E^{s}_{\lambda}f(x)=\sum_{\lambda_k<\lambda}\left(1-\frac{\lambda_k}{\lambda}\right)(f,u_k)u_k(x) \] are convergent.