One more (a further) generalization of the Final Value Theorem

TL;DR

This paper extends the Final Value Theorem to higher-order running averages, allowing for the calculation of average values of functions under broader conditions than previously possible.

Contribution

It introduces a new generalization of the Final Value Theorem applicable to higher-order running averages with specific existence conditions.

Findings

Extended the theorem to higher-order averages

Established conditions for the existence of the limits

Broadened applicability of the Final Value Theorem

Abstract

It is shown that the previous [1-3] generalization of the final-value theorem to the average (not necessarily limiting) values, can be extended to the higher-order running averages $(<>_{t}), lim_{s\rightarrow0}[sF(s)] = lim_{t\rightarrow\infty} << ... << f(\lambda_{q}) > \lambda_{q-1} > \lambda_{q-2} ... > \lambda_{1} > t$, if f(t) is such that $\exists m = min[q] < \infty$ for which the later limit exists.