Sequences of linear operators related to Cesàro-convergent sequences

Abstract

Given a Cesàro-convergent sequence of real numbers\((a_{n})_{n\in \mathbb{N}}\), a sequence \((\varphi_{n})_{n\in\mathbb{N}}\) of operators is defined on the Banach space \(\mathcal{R}(I,F)\) of regular functions defined on \(I=[0,1]\) and having values in a Banach space \(F\), \[\varphi_{n}(f)=\frac{1}{n}\sum_{k=1}^{n}a_{k}f\left( \tfrac{k}{n}\right) .\]It is proved that if, in addition, the sequence \(\big( \frac{\left|a_{1}\right| +\ldots+\left| a_{n}\right| }{n}\big)_{n\in\mathbb{N}}\) is bounded, then \(\varphi_{n}(f)\) converges to \(a\cdot\int\nolimits_{0}^{1}f,\) where \(a=\lim_{n\rightarrow\infty}\frac{a_{1}+\ldots+a_{n}}{n}.\) The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the space \(\mathcal{C}^{1}(I,F)\).