Whittaker-Kotelnikov-Shannon sampling theorem and aliasing error

Let B-sigma,B-p, 1 less than or equal to p less than or equal to infinity, be the set of all functions from L(p)(R) which can be continued to entire functions of exponential type less than or equal to sigma. The well known Whittaker-Kotelnikov-Shannon sampling theorem states that every f is an element of B-sigma,B-2 can be represented as f(x)=k is an element of Z Sigma f(k pi/sigma) sin sigma(x-k pi/sigma)/sigma(x-k pi/sigma), sigma>0, in norm L(2)(R). We prove that it is also true for all f is an element of B-sigma,B-p,B- 1<p<infinity, in norm L(p)(R). From this, we further prove that if f(x) = O(psi(x)), where psi(x)is an element of L(p)(R), psi(x)greater than or equal to 0 is even and non-increasing on [0, infinity), and f(x) is Riemann integrable on every finite interval, then the aliasing error of f; i.e., f(x)-Sigma(k is an element of Z) f(k pi/sigma sin sigma(x-k pi/sigma)[sigma(x-k pi/sigma)](-1), converges to zero in L(p)(R), 1<p<infinity, when sigma--> + infinity. If f is an element of L(p)(r)(R), r is an element of N, we also determine the error bound of its aliasing error. (C) 1996 Academic Press, Inc.