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Riesz mean
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.
==Definition==
Given a series $\$, the Riesz mean of the series is defined by
:$s^\delta\left(\lambda\right) =$
\sum_ \left(1-\frac\right)^\delta s_n
Sometimes, a generalized Riesz mean is defined as
:$R_n = \frac \sum_^n \left(\lambda_k-\lambda_\right)^\delta s_k$
Here, the $\lambda_n$ are sequence with $\lambda_n\to\infty$ and with $\lambda_/\lambda_n\to 1$ as $n\to\infty$. Other than this, the $\lambda_n$ are otherwise taken as arbitrary.
Riesz means are often used to explore the summability of sequences; typical summability theorems discuss the case of $s_n = \sum_^n a_k$ for some sequence $\$. Typically, a sequence is summable when the limit $\lim_ R_n$ exists, or the limit $\lim_s^\delta\left(\lambda\right)$ exists, although the precise summability theorems in question often impose additional conditions.

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