Polynomial function and derivative approximation of Sinc data

Sinc methods consist of a family of one dimensional approximation procedures for approximating nearly every operation of calculus. These approximation procedures are obtainable via operations on Sinc interpolation formulas. Nearly all of these approximations–except that of differentiation–yield exceptional accuracy. The exception: when differentiating a Sinc interpolation formula that gives an approximation over an interval with a finite end-point. In such cases, we obtain poor accuracy in the neighborhood of the finite end-point. In this paper we derive novel polynomial-like procedures for differentiating a function that is known at Sinc points, to obtain an approximation of the derivative of the function that is uniformly accurate on the whole interval, finite or infinite, in the case when the function itself has a derivative on the closed interval.