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

## TL;DR

This paper develops a rational approximation of the sinc function using sampling and Fourier transforms, generalizing previous cosine product identities, with a MATLAB implementation demonstrating high accuracy.

## Contribution

It introduces a generalized approach for rational approximation of the sinc function based on sampling and Fourier transforms, extending prior cosine product identities.

## Key findings

- Achieves high-precision approximation with 16 terms
- Provides a MATLAB code for validation
- Extends previous cosine product-to-sum identities

## Abstract

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.