Sharp results on sampling with derivatives in bandlimited functions

Abstract: We discuss the problems of uniqueness, sampling and reconstruction with derivatives in the space of bandlimited functions. We prove that if X is sequence of real numbers such that the maximum gap between two consecutive samples is less than certain positive constant c, then bandlimited function of bandwidth {\sigma} can be reconstructed uniquely and stably from its nonuniform samples involving derivatives. We also prove that if the maximum gap is less than or equal to c, then X is a set of uniqueness for the space of bandlimited functions of bandwidth {\sigma} when the samples involving k-1 derivatives. As a by-product we obtain the sharp maximum gap condition for samples involving first derivatives.