One Dimensional Sinc Theory
This chapter contains derivations of one dimensional Sinc approximations. Whereas previous derivations of these results were done using complex variables, the majority of the derivations in this section are done using Fourier transforms. We thus present derivations of Sinc approximation, over a ﬁnite interval (a, b) , over a semi-inﬁnite interval (0,∞) , over the whole real line (−∞ ,∞), and more generally over arcs in the complex plane. Sinc approximations are provided for the following types of operations:
Interpolation; Approximation of derivatives; Quadrature; Indeﬁnite integration; Indeﬁnite convolution;
Fourier transforms; Hilbert transforms; Cauchy transforms; Analytic continuation; Laplace transforms, and their inversion; Solutions of Volterra integral equations; Solutions of Wiener Hopf integral equations; and Solutions of ordinary diﬀerential equation initial value problems.
Let me make a few general remarks about Sinc methods.