# Given a multivariate rational function p ( <mrow class="MJX-TeXAtom-ORD"> <mover>

Given a multivariate rational function $p\left(\stackrel{\to }{x}\right)=\frac{f\left(\stackrel{\to }{x}\right)}{g\left(\stackrel{\to }{x}\right)}$ over $\left[0,1{\right]}^{n}$ with $p\left(\stackrel{\to }{x}\right)\in \left[0,1\right]$, how can we come up with a polynomial approximation of $p$, say $q\left(\stackrel{\to }{x}\right)$ such that $|p\left(\stackrel{\to }{x}\right)-q\left(\stackrel{\to }{x}\right)|\le ϵ$ for all $\stackrel{\to }{x}$ ?
For the univariate case, we might use chebyshev approximation, however, what are the results for the multivariate case?
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Tanner Hamilton
Approximation of continuous functions of several variables on $\left[0,1{\right]}^{n}$ can be done by means of multivariate Chebyshev polynomials that are the tensor product of Chebyshev polynomials in one variable. For $f\in C\left(\left[0,1{\right]}^{n}\right)$,
$f\left({x}_{1},\dots ,{x}_{n}\right)\approx \sum _{1\le {k}_{i}\le N}{a}_{{k}_{1},\dots ,{k}_{n}}{T}_{{k}_{1}}\left({x}_{1}\right)\cdots {T}_{{k}_{n}}\left({x}_{n}\right).$