A DISTRIBUTIONAL SAMPLING THEOREM

The classical Shannon sampling theorem has many extentions [ii Tutoria l in Theory and Applications, C. K. Chui, ed., Academic Press, 1992, p p. 51-70], one of which is to functions of polynomial growth. In this paper, we shall prove the following theorem: Let F(omega) be a distrib ution with compact support on [-pi,pi] and f(t) be the Fourier transfo rm. Then f(t) = (n) Sigma f(n) sin pi(t - n)/pi(t - n) holds in the se nse of (C, cu) summation under a very mild condition. The above result improves the theorems in both [SIAM J. Math. Anal., 19 (1988), pp. 11 98-1203] and [Generalized Functions, Convergence Structures, and Their Applications, B. Stankovic et al., eds., Plenum Press, 1988, pp. 349- 357] given by G. G. Waiter.