Riesz summation method

A method for summing series of numbers and functions; denoted by $(R,\lambda,k)$. A series $\sum_{n=0}^\infty a_n$ is summable by the Riesz summation method $(R,\lambda,k)$ to the sum $s$ if

$$\lim_{\omega\to+\infty}\sum_{\lambda_n\leq\omega}\left(1-\frac{\lambda_n}{\omega}\right)^ka_n=s,$$

where $k>0$, $0\leq\lambda_0<\ldots<\lambda_n\to\infty$, and $\omega$ is a continuous parameter. The method was introduced by M. Riesz [1] for the summation of Dirichlet series. The method $(R,\lambda,k)$ is regular; when $\lambda_n=n$ it is equivalent to the Cesàro summation method $(C,k)$ (cf. Cesàro summation methods), and these methods are compatible (cf. Compatibility of summation methods).

Riesz considered also a method in which summability of the series $\sum_{n=0}^\infty a_n$ is defined by means of the limit of the sequence $\{\sigma_m\}$, where

$$\sigma_m=\frac{1}{P_m}\sum_{k=0}^mp_ks_k,$$

$$P_m=\sum_{k=0}^mp_k\neq0,\quad s_k=\sum_{n=0}^ka_n.$$

This method is denoted by $(R,p_n)$. The method $(R,\lambda,k)$ is a modification of the method $(R,p_n)$ (when $k=1$) and is a generalization of it to the case of an arbitrary $k>0$.

References

 [1] M. Riesz, "Une méthode de sommation équivalente à la méthode des moyennes arithmétique" C.R. Acad. Sci. Paris , 152 (1911) pp. 1651–1654 [2] F. Riesz, "Sur la sommation des séries de Dirichlet" C.R. Acad. Sci. Paris , 149 (1909) pp. 18–21 [3] G.H. Hardy, M. Riesz, "The general theory of Dirichlet series" , Cambridge Univ. Press (1915) [4] G.H. Hardy, "Divergent series" , Clarendon Press (1949)