Error estimates for sampling sums based on convolution integrals

The classical Shannon sampling theorem is concerned with the representation of bandlimited signal functions by a sum built up from a countable number of samples. It is shown that a not necessarily bandlimited function f can be approximately represented by generalized sampling sums which originate from discretized convolution integrals known, e.g., in approximation theory. The rate of convergence of the new sums to f is precisely as good as that of the associated convolution integrals. This gives sufficient as well as matching necessary conditions for a certain rate of convergence