Absolute Riesz summability factors (Q5903035)

Suppose that \(\{\lambda_ n\}\) is a sequence of positive unboundedly increasing real numbers with \(\lambda_ 1>0\) and that \(k\geq 0\). Let \(R^ k_{\lambda}(w)=w^{-k}\sum_{\lambda_{\nu}<w}(w- \lambda_{\nu})^ ka_{\nu}\) for \(w>\lambda_ 1\) and zero otherwise. If \(R^ k_{\lambda}(w)\) is of bounded variation with respect to w on the interval (h,\(\infty)\) where \(h\geq \lambda_ 1\), then we say that \(\Sigma a_{\nu}\) is absolutely Riesz summable of type \(\lambda\) and order k, or summable \(| R,\lambda_{\nu},k|\). If \(k>0\), \(m\geq 1\), \(k>1-1/m\) and \(\int^{\infty}_{\lambda_ 1}w^{m-1}| (d/dw)R^ k_{\lambda}(w)|^ mdw<\infty\), then we say that \(\sum a_{\nu}\) is summable \(| R,\lambda_{\nu},k|_ m\). The authors prove the following result: Theorem. If \(k>0\), if \(\eta >0\), if \(m>1\), if \(\int^{X}_{\lambda_ 1}w^{m-1-\eta}| (d/dw)R^ k_{\lambda}(w)|^ mdw=O(\Lambda (X))\) as \(X\to \infty\), where \(\Lambda\) (X) is positive monotone non-decreasing for \(X\geq \lambda_ 1\) and if there exists a function h(u)\(\in L(0,x)\) for every finite x such that \[ \epsilon_{\nu}=\int^{\infty}_{\lambda_ 1}(u- \lambda_{\nu})^ kh(u)du\quad with\quad \int^{\infty}_{\lambda_ 1}u^{km+m+\eta -1}\Lambda (u)| h(u)|^ mdu<\infty, \] then \(\sum a_ n\epsilon_ n\) is summable \(| R,\lambda_ n,k|_ m\). The authors state conditions under which their result is best possible. However, they do not give any examples of series \(\sum a_ n\) and factor sequences \(\{\epsilon_ n\}\) that satisfy the theorem.