The use of the sinc-Gaussian sampling formula for approximating the derivatives of analytic functions

The sinc-Gaussian sampling formula is used to approximate an analytic function, which satisfies a growth condition, using only finite samples of the function. The error of the sinc-Gaussian sampling formula decreases exponentially with respect to N, i.e., N(-1/2)e(-N), where is a positive number. In this paper, we extend this formula to allow the approximation of derivatives of any order of a function from two classes of analytic functions using only finite samples of the function itself. The theoretical error analysis is established based on a complex analytic approach; the convergence rate is also of exponential type. The estimate of Tanaka et al. (Jpan J. Ind. Appl. Math. 25, 209-231 2008) can be derived from ours as an immediate corollary. Various illustrative examples are presented, which show a good agreement with our theoretical analysis.