Riesz mean

Remarks

Example: take $\lambda_n = n$, $\delta = 1$ and an integer cut-off $\omega = N + 1$. Then $$R^1_{\lambda}(N + 1) = \sum_{n=0}^{N} \frac{N + 1 - n}{N + 1} a_n = \frac{s_0 + s_1 + \cdots + s_N}{N + 1},$$ where $s_j$ are the partial sums, because $a_n$ occurs in exactly the $N + 1 - n$ partial sums $s_n, \dots, s_N$. So these Riesz means are the Cesàro means of the partial sums.

Non-example: if $\lambda_n = 1 - 2^{-n}$, the sequence is increasing but bounded, so the condition $\lambda_n \to \infty$ fails. Then for $\omega > 1$ every term enters $R^{\delta}_{\lambda}(\omega)$, which is not a finite sum, and the definition does not apply.

For Dirichlet series $\sum a_n n^{-s}$ one usually takes $\lambda_n = \ln n$ for $n \ge 1$.