Around Taylor's theorem on the convergence of sequences of functions

Rok, strany: 2021, 129 - 138

Egoroff's classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor's theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable fnctions $\{f_n\}_{n\in\mathbb N}$ pointwise converging to a function $f$ on a measurable set $E$, there exist a decreasing sequence $\{δ_n\}_{n\in\mathbb N}$ of positive reals converging to $0$ and a set $A\subseteq E$ such that $E\setminus A$ is a nullset and $\lim_n\to+∞((|f_n(x)-f(x)|) / (δ_n))=0$ for all $x\in A$. Let $J(A,\{f_n\})$ denote the set of all such sequences $\{δ_n\}_{n\in\mathbb N}$. The main results of the paper concern basic properties of sets of all such sequences for a given set $A$ and a given sequence of functions. A relationship between pointwise convergence, uniform convergence, and Taylor's type of convergence is considered.

ISO 690:

Horbaczewska, G., Rychlewicz, P. 2021. Around Taylor's theorem on the convergence of sequences of functions. In Tatra Mountains Mathematical Publications, vol. 78, no.1, pp. 129-138. 1210-3195. DOI: https://doi.org/ 10.2478/tmmp-2021-0009

APA:

Horbaczewska, G., Rychlewicz, P. (2021). Around Taylor's theorem on the convergence of sequences of functions. Tatra Mountains Mathematical Publications, 78(1), 129-138. 1210-3195. DOI: https://doi.org/ 10.2478/tmmp-2021-0009

Vydavateľ: Mathematical Institute, Slovak Academy of Sciences, Bratislava

https://creativecommons.org/licenses/by-nc-nd/4.0/