General sampling theorem using contour integral ✩

We present the sampling theorem with sampling functions of general form for entire functions satisfying one of the growth conditions 1 + |y| f (z) A 1 + |z| N1 exp τ |x| + σ |y| , 1 + |x| f (z) A 1 + |z| N1 exp τ |x| + σ |y| for some A > 0, τ,σ 0, N1 ∈ N ∪ {0} and any z = x + iy ∈ C. It will be shown that many well-known sampling theorems included in SIAM J. Math. Anal. 19 (1988) 1198–1203 and Inform. Control 8 (1965) 143–158 can be interpreted as special cases of this sampling theorem. As examples, we provide sampling representations for entire functions which are bounded, of polynomial growth, or of exponential growth on R.We also provide sampling representations involving derivatives of entire functions and nonuniform sampling representations. Taking the set of sampling points in which a finite number of points are arbitrarily distributed, we obtain a sampling representation.  2003 Elsevier Inc. All rights reserved.