A rational approximation of the sinc function based on sampling and the Fourier transforms

Abstract

In our previous publications we have introduced the cosine product-to-sum identity [17] $$\prod _{m=1}^M\cos \left(\frac{t}{2^m}\right)=\frac{1}{2^{M-1}}\sum _{m=1}^{2^{M-1}}\cos \left(\frac{2m-1}{2^M}t\right)$$ and applied it for sampling [1, 2] as an incomplete cosine expansion of the sinc function in order to obtain a rational approximation of the Voigt/complex error function that with only $16$ summation terms can provide accuracy $\sim 10^{-14}$. In this work we generalize this approach and show as an example how a rational approximation of the sinc function can be derived. A MATLAB code validating these results is presented.

20 pages, 7 figures