A Composition Theorem for Binomially Weighted Averages

Abstract

We study binomially weighted summation methods given by \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\binom{n}{k}r^k(1-r)^{n-k}x_k\right)_{n\in \mathbb{N}} \] for $r\in (0,1)$, and their behavior under composition with summation methods of the form \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^nλ_k x_{n-k}\right)_{n\in \mathbb{N}}. \] Our main result shows that if the binomially weighted averages of a sequence $(x_n{)}_{n\in \mathbb{N}}$ converge to a limit then the binomially weighted averages of the sequence ${\left({\sum }_{k=0}^n\hat{I}{»}_k{x}_{n-k}\right)}_{n\in \mathbb{N}}$ converge to the same limit whenever $(\hat{I}{»}_n{)}_{n\in \mathbb{N}}$ is an absolutely summable sequence with ${\sum }_{k=0}^{\infty }\hat{I}{»}_k=1$. This result disproves a theorem appearing in the literature. Additionally, we discuss applications and extensions of our main result to compositions with weighted Cesà ro averages.