Sampling procedures in function spaces and asymptotic equivalence with shannon's sampling theory

Abstract

We view Shannon's sampling procedure as a problem of approximation in the space S = {s: s (x) = (c * sinc)(x)c ε l ₂}. We show that under suitable conditions on a generating function λ ε L ₂, the approximation problem onto the space V = {v:v(x) = (c * λ)(x)c ε l ₂} produces a sampling procedure similar to the classical one. It consists of an optimal prefiltering, a pure jitter-stable sampling, and a postfiltering for the reconstruction. We describe equivalent signal representations using generic, dual, cardinal, and orthogonal basis functions and give the expression of the corresponding filters. We then consider sequences λⁿ, where λⁿ denotes the n-fold convolution of λ. They provide a sequence of increasingly regular sampling schemes as the value of n increases. We show that the cardinal and orthogonal pre- and postfilters associated with these sequences asymptotically converge to the ideal lowpass filter of Shannon. The theory is illustrated using several eamples.