On stable summation of Fourier series (Q735818)

The author studies the Fourier expansion in a Hilbert space with a complete orthonormal system of elements \(\{\varphi_k\}_{k=1}^{\infty}\) and using a sequence of positive real numbers \(\{\lambda_k\}_{k=1}^{\infty}\) with \[ \lambda_k\geq \lambda_{k+1}\;(k\geq 1),\;\;\lim_{n\rightarrow\infty}\lambda_n=0, \] to define a self-adjoint completely continuous operator with \[ Ax=\sum_1^{\infty}\,\lambda_k (x,\varphi_k)\,\varphi_k. \] (as all \(\lambda_k\) are different from zero, ker\(A=\{0\}\)) \vskip0.2cm Two theorems are stated, the first preceded by the sentence ``The assertion below follows directly from [4, p. 23]'' and the second by ``The result below follows from Theorem 4 in [4, p. 30]'', without any further proof. Here [4] is the work \textit{Iterative methods for solving ill-posed operator equations with smooth operators} by \textit{A.B. Bakushinskii} and \textit{M. Yu. Kokorin} [Editorial URSS, Moscow (2002); in Russian].