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

In our previous publications we have introduced the cosine product-to-sum identity [17] $\prod\limits_{m = 1}^M {\cos \left( {\frac{t}{2^m}} \right)} = \frac{1}{2^{M - 1}}\sum\limits_{m = 1}^{2^{M - 1}} {\cos \left( {\frac{2m - 1}{2^M}t} \right)}$ 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.

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