Efficiently approximating functions

Info

This note is entirely based on the more comprehensive Chebyshev Approximation and How It Can Help You Save Money, Win Friends, and Influence People

The problem we want to solve is how to efficiently approximate functions as hardware, for when speed is essential. An example would be the approximation of trigonometric functions, frequently used, as specific chips.

The suggested solution relies on doing a Taylor series expansion of a function $f(x)$ around a point $x_0$. The choice of point $x_0$ and coefficients depends on the range $[a,b]$ over which we want to approximate the function, since the errors rise as we get further away from $x_0$. The recommendation is to

1. Make a linear transformation $x\to u$ such that the range of interest is $[-1,1]$ when expressed as a function of $u$
2. Calculate the coefficients of the Chebyshev Polynomials by taking $c_n={\int }_{-1}^{1}f(u)T_n(u)du$, i.e. the Inner Product over the interval $[-1,1]$. This can be better achieved in Python using numpy’s chebfit.