Sampling sums with kernels of finite oscillation

The paper deals with so-called generalized sampling sums stemming from the well-known Shannon sampling series; here the sine-function is replaced by suitable time limited even kernel functions. It is shown that the best possible rate of approximation for such processes to functions of class C(IR) is c)(w&-2m-2), where 2m is the number of sign changes of the kernel and the sampling rate is w-1. Furthermore, a general method for the construction of linear combinations-of positive kernels leading to such approximation rates is presented.