Korovkin Type Results for Multivariate Continuous Periodic Functions

Abstract

Let k be a natural number and $C_{2\pi }\left( \mathbb {R}^{k}\right)$ the real Banach space of the all continuous $2\pi$-periodic functions $f: \mathbb {R}^{k}\rightarrow \mathbb {R}$. We prove the following Korovkin-type result: Let $\gamma :\mathbb {R}^{k}\times \mathbb {R}^{k}\rightarrow \left[ 0,\infty \right)$ be a continuous trigonometric separating function which is $2\pi$-periodic with respect to every variable and such that $\gamma \left( t,t\right) =0$ for all $t\in \mathbb {R}^{k}$. If $V_{n}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow C_{2\pi }\left( \mathbb {R} ^{k}\right)$ is a sequence of positive linear operators such that $\lim \nolimits _{n\rightarrow \infty }V_{n}\left( \gamma \left( \cdot ,t\right) \right) \left( t\right) =0$ uniformly with respect to $t\in \mathbb {R}^{k}$ and $\lim \nolimits _{n\rightarrow \infty }V_{n}\left( \mathbf {1}\right) = \mathbf {1}$ uniformly on $\mathbb {R}^{k}$ then, $\lim \nolimits _{n\rightarrow \infty }V_{n}\left( f\right) =f$ uniformly on $\mathbb {R}^{k}$ for every $f\in C_{2\pi }\left( \mathbb {R}^{k}\right)$. As an application we obtain the Korovkin theorem for positive linear operators on the space $C_{2\pi }\left( \mathbb {R}^{k}\right)$. Various applications are given.