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Given a continuous function $f:[a,b]\rightarrow\mathbb{R}$, it is a well known result that, for each $n\geq0$, there exists a polynomial $p_n$ that best approximates $f$ in $\|\cdot\|_\infty$ among all polynomials of degree less than or equal to $n$. Moreover, if $f$ is analytic, then $p_n\rightarrow f$ at an exponential rate.

My question is whether these results generalize to several variables. Let $I=[a_1,b_1]\times\cdots \times[a_n,b_n]$ be a multivariate rectangle and $f:I\rightarrow\mathbb{R}$ be a continuous function. Let $p_n$ be the best approximation multivariate polynomial, for each degree $n\geq0$. If $f$ is analytic, then $p_n\rightarrow f$ at an exponential rate. A reference on this result would be enough for me.

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