Suppose $f(x_1,x_2)$ is a polynomial of degree $(k_1,k_2)$. Consider it on $[0,1]\times[0,1]$.

Let $B_{L_1,L_2}(f)$ denote the bivariate Bernstein polynomial of $f$ for $L_1 \geq k_1$ and $L_2 \geq k_2$.

I was wondering if somebody could help and indicate a reference where it is shown that $B_{L_1,L_2}(f)$ is in fact a polynomial of degree $(k_1,k_2)$.

Update: I found the proof of an analogous result for univariate polynomials in Natanson "Constructive Function Theory." Will try to generalize it to the case of bivariate polynomials but it would be easier of there was a reference available for a polynomial of several variables.


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