# How to find coefficients in a multivariate Chebyshev polynomial approximation

How do perform a multivariate Chebyshev approximation?

Let \begin{align} \vec{x} & = x_{0}, x_{1}, ... , x_{n},\\ \vec{a} & = a_{0}, a_{1}, ... , a_{n},\\ \vec{b} & = b_{0}, b_{1}, ... , b_{n}. \end{align}

Let $f(\vec{x})$ be an arbitrary multivariate function.

What is the correct form for Chebyshev polynomials and coefficients which approximate an arbitrary multivariate function $f(\vec{x})$ on the domain $a_{i} < x_{i}< b_{i} \forall i \in [0,n]$?