The Shannon Sampling Series and the Reconstruction of Signals in Terms of Linear, Quadratic and Cubic Splines

The sine-kernel function of the sampling series is replaced by spline functions having compact support, all built up from the B-splines $M_{n} $. The resulting generalized sampling series reduces to finite sums so that no truncation error occurs. Moreover, the approximation error generally decreases more rapidly than for the classical series when W tends to infinity, $1/W$ being the distance between the sampling points. For the kernel function $\varphi (t) = 5M_4 (t) - 4M_5 (t)$ with support $[ - \tfrac{5}{2},\tfrac{5}{2} ]$, e.g., it decreases with order $O(W^{ - 4} )$ provided the signal has a fourth order derivative; for the classical series the order is $O(W^{-4}\log W)$. Seven characteristic examples are treated in detail.