On the equivalence of the classical methods of summation of orthogonal series (Q1812550)

Let \(\{\varphi_k\}^\infty_{k=0}\) be an arbitrary orthonormal system of functions in a measure space \((X,{\mathcal F},\theta)\). Also, let \(S_n=\sum^n_{k=0} a_k\varphi_k\), \(\sigma^ \alpha_n = \sum^n_{k=0} (A^{\alpha-1}_{n=k}/A^\alpha_n)S_k\) \((\alpha>0)\), \(n=1,2,\dots\), and \(A_ r=\sum^ \infty_{k=0} r^k a_k\varphi_k\), \(0\le r<1\), denote the partial sums, Cesàro means and Abel means, respectively, of the series (1) \(\sum^ \infty_{k=0} a_k\varphi_k\) (with \(\sum^ \infty_{k=0} a^2_k<\infty)\), which converges in the norm of \(L^2(X)\). \textit{S. Kaczmarz} [Math. Ann. 96, 148--151 (1927; JFM 52.0276.01)] and \textit{A. Zygmund} [Fundam. Math. 10, 356--362 (1927; JFM 53.0267.04)] established that for the orthogonal series (1) of class \(L^2(X)\) all Cesàro summation methods of order \(\alpha>0\) and the Abel method are equivalent to each other and to the convergence of the subsequence of partial sums with indices \(2^n\), \(n\ge 0\) (i.e., if the function \(F\) is an \(L^2(X)\)-sum of series (1), then, for given \(\alpha>\beta>0\), \[ \lim_{n\to\infty}\sigma_n^\alpha = \lim_{n\to\infty}\sigma_n^\beta = \lim_{r\to 1-0}A_r = \lim_{n\to\infty}S_{2^n} = F \] almost everywhere on an \({\mathcal F}\)-measurable subspace \(Y_{\alpha,\beta}\subseteq X\), for which any of the four limits does not exist at almost every point of the complement \(X\backslash Y_{\alpha,\beta})\). Following from this, \textit{O. A. Ziza} [Mat. Sb., Nov. Ser. 66(108), 354--377 (1965; Zbl 0154.06205)] proved that for series of the form (1) Euler summation methods (of positive order) and the Borel method are equivalent to the convergence of the subsequence of the partial sums \(S_{n^ 2}\), \(n\geq 0\). On the basis of a construction in the above-mentioned works of \textit{Kaczmarz} [loc. cit.] and \textit{Zygmund} [loc. cit.], the author simplifies to some extent the proof of Ziza's theorem and also shows that for any series of form (1) the upper (or lower) limits on indeterminacy for Euler means, Borel sums and the partial sums \(S_{n^2}\), \(n\geq 0\), coincide on \(X\) (with the exception of a set of zero \(\theta\)-measure), and, further, in a similar way, \[ \limsup_{n\to\infty}\sigma_n^\alpha=\limsup_{r\to 1-0}A_r = \limsup_{n\to \infty} S_{2^n} \] almost everywhere on \(X\) for all \(\alpha>0\). In a parallel manner the author makes additional statements.