On approximation by Riesz means concerning orthogonal series (Q1842728)

For an orthonormal system \(\{\phi_n(x)\}\) on an interval \((a, b)\), we consider a real orthogonal series \(\sum^\infty_{n= 0} c_n \phi_n(x)\) with \(\sum^\infty_{n= 0} c^2_n< \infty\), together with its partial sum function \(s_\omega(x):= \sum_{k\leq \omega} c_k\phi_k(x)\) \((\omega\geq 0)\). Then \((s_n(x))\) converges in the \(L^2\) norm to a square-integrable function \(f(x)\). Given a non-negative strictly increasing function \(\lambda(\omega)\) \((\omega\geq 0)\) such that \(\lambda(0)= 0\), \(\lambda(n+ 1)\leq M\lambda(n)\) \((M\geq 1,\;n= 1,2,\dots)\), \(\lambda(n)\to \infty\), the \((R, \lambda(n), 1)\)-means of the series are \(R_\omega(x):= [\lambda(\omega)]^{- 1} \sum_{k\leq \omega} [\lambda(\omega)- \lambda(k)] c_k \phi_k(x)\) and \(\sum c_n \phi_n(x)\) is \((R, \lambda(n), 1)\)-summable to \(f(x)\) when \(R_\omega(x)\) \(\to f(x)\) \((\omega\to \infty)\). Sufficient conditions are given (Theorem A\(^*\)) in order that the Riesz means should approximate the function to a certain order, namely that, for a positive non-decreasing function \(\ell(\omega)\), we have \(|R_k(x)- f(x)|= o_x(\ell(k)^{- 1})\) almost everywhere in a measurable set \(E\subseteq (a, b)\). There are two related theorems (B\(^*\) and C\(^*\)). These all use the concept of a quasi-geometrically monotone sequence to generalize corresponding Theorems A, B, C which appeared in a previous paper of the author.