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Strong summability of Riesz means. (English. Russian original) Zbl 0656.42016
Math. Notes 40, 706-713 (1986); translation from Mat Zametki 40, No. 3, 352-363 (1986).
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).
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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