Approximation of functions and their derivatives by generalized sampling series

Abstract

For any bounded function $f(t)$ defined on $\mathbb{R}$ and continuous at $t \in \mathbb{R}$, we consider a generalized sampling series given by $(S_{W}^{\varphi}f) (t) := \sum_{k\in \mathbb{Z}} f(\frac{k}{W}) \varphi(Wt-k),~~~(t \in \mathbb{R} ;W>0)$. we find sufficient conditions on the reconstruction function $\varphi(t)$, under which we have $\partial_t^{n} (S_{W}^{\varphi}f) (t) = \sum_{k\in \mathbb{Z}} f(\frac{k}{W}) \partial_t^{n} \varphi(Wt-k)$ converges to $f^{(n)}(t)$ as $W \rightarrow \infty$.