A Composition Theorem for Binomially Weighted Averages

TL;DR

This paper proves a composition theorem for binomially weighted averages, showing convergence preservation under certain conditions, and discusses related applications and extensions.

Contribution

It establishes a new composition theorem for binomially weighted averages, correcting a previous misconception and extending to weighted Cesàro averages.

Findings

Convergence of binomially weighted averages is preserved under composition with absolutely summable sequences summing to one.

Disproves a previously published theorem in the literature.

Extends the main result to weighted Cesàro 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 \[ (x_n)_{n\in \mathbb{N}} \mapsto \left(\sum_{k=0}^n\lambda_k x_{n-k}\right)_{n\in \mathbb{N}}. \] 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 $\left(\sum_{k=0}^n\lambda_kx_{n-k}\right)_{n\in \mathbb{N}}$ converge to the same limit whenever $(\lambda_n)_{n\in\mathbb{N}}$ is an absolutely summable sequence with $\sum_{k=0}^{\infty}\lambda_k = 1$. This result disproves a theorem appearing in the literature. Additionally, we discuss applications and extensions of our main result to compositions with weighted Ces\`aro averages.