A theorem concerning absolute Euler summability of orthonormal series (Q1074798)

The Euler means of order q, \(q>0\) of a sequence \((S_ n(x))\) are defined by the formula \[ E^ q_ n(x)=\frac{1}{(1+q)^ n}\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)q^{n-k}\quad S_ k(x)\quad n=0,1,2,.... \] Denoting \(S_ n(x)=\sum^{n}_{k=0}c_ k\phi_ k(x)\) of the nth partial sum of the real orthonormal series \(\sum^{\infty}_{n=0}c_ n\phi_ n(x)\) with respect to the orthonormal system \((\phi_ n)\) in \(<0,1>\). The author proves the following theorem: Let \(\beta <5/2\), \(0<r\leq 2\) and \(r>\frac{4}{7}(\beta +1)\). Let \(H_ n=(\sum^{2^{n+1}}_{k=2^ n+1}k^{(e/7)}(\beta +1)^{-3/2} c^ 2_ k\quad)^{r/2}\) of \(\sum^{\infty}_{n=0}H_ n<\infty\), then \(\sum^{\infty}_{n=0}n^{\beta} | E^ q_ n(x)- E^ q_{n-1}| <\infty\) almost everywhere in \(<0,1>\). This theorem generalizes the results of \textit{O. A. Ziza} [Mat. Sb., Nov. Ser. 66(108), 354-377 (1965; Zbl 0154.062)] and \textit{V. N. Spevakov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1981, No.2(225), 70-73 (1981; Zbl 0469.42005)] in the sense that for \(\beta =1/2\), \(r=2\) it reduces to the theorem of Ziza and for \(\beta =0\), \(r=1\) it reduces to the theorem of Spevakov.