Consider the multivariate case for the question "Approximation theory reference for a bounded polynomial having bounded coefficients" ([Approximation theory reference for a bounded polynomial having bounded coefficients](https://mathoverflow.net/questions/97769/approximation-theory-reference-for-a-bounded-polynomial-having-bounded-coefficie))


I tried to find an upper bound for coefficients for each monomial. I tried two methods:


1. Lagrange interpolation. As in Lemma 4.1 in <http://eccc.hpi-web.de/report/2012/037/download> .
2. I first only consider the variable $x\_1$ and get an upper bound for the coefficients. Now the bound is the upper bound for another polynomial of $x\_2,x\_3,...,x\_n$ and we can repete the above method.


However, using either method above, the upper bound will contain factor $d^n$ or $c^{d\*min\{d,n\}}$ where c is a constant, which I think is too large. Is it possible to find a better upper bound?



A bounded polynomial having bounded coefficients: several variables
Bounds in the univariate case, see e.g. [here](https://mathoverflow.net/questions/257909/find-the-maximum-of-a-p-if-a-0a-1x-dotsa-nxn-1-1-mapsto-1-1/257918#257918), were established by V.A. Markov in 1892. 


S.N. Bernstein has given an extension of the result to the multivariate case in 



> 
> S.N. Bernstein, On certain elementary extremal properties of
>  polynomials in several variables, Doklady Akad. Nauk SSSR (N.S.) 59
>  (1948), 833-836.
> 
> 
> 


Unfortunately I do not have access to that paper, but according to MR0023953, the following result is proved. Let
$$P(x\_{1},\ldots,x\_{n})=\sum\_{1\leq\alpha\_{h}\leq d\_{h}} A\_{\alpha\_{1},\ldots,\alpha\_{n}}x\_{1}^{\alpha\_{1}}\ldots
x\_{n}^{\alpha\_{n}},$$
such that $|P|\leq1$ on the cube $|x\_{h}|\leq1$, $1\leq h\leq n$. Then
$$|A\_{\alpha\_{1},\ldots,\alpha\_{n}}|\leq\frac{\prod\_{h=1}^{n}|B\_{\alpha\_{h}}^{(d\_{h})}|}{\alpha\_{1}!\ldots\alpha\_{n}!},$$
where $B\_{\alpha\_{h}}^{(d\_{h})}$ is the coefficient of $x^{\alpha\_{h}}/\alpha\_{h}!$ in the expansion of the Chebyshev polynomial
$$T\_{m}(x)=\cos(m\cos^{-1}(x))=\sum\_{i=0}^{m}B\_{i}^{(m)}x^{i}/i!,$$
and $m=d\_{h}$ if $d\_{h}-\alpha\_{h}$ is even and $m=d\_{h}-1$ if $d\_{h}-\alpha\_{h}$ is odd.


For the particular case of homogeneous polynomials 
$$P\_{d}(x\_{1},\ldots,x\_{n})=\sum\_{|\alpha|= d}c\_{\alpha}x^{\alpha},$$ 
of total degree $d$, where $\alpha$ are multi-indices, such that
$|P\_{d}|\leq1$ on the ball $x\_{1}^{2}+\cdots +x\_{n}^{2}\leq1$, upper bounds on the coefficients are given in Theorem 2 of



> 
> O.D. Kellogg, On bounded polynomials in several variables, Math. Z.
>  27 (1928), no. 1, 5-64.
> 
> 
> 


namely, the coefficient of the monomial $x\_{1}^{k\_{1}}\ldots x\_{n}^{k\_{n}}$ cannot exceed in absolute value
$$\frac{d!}{k\_{1}!\ldots k\_{n}!}.$$


For general (nonhomogeneous) polynomials of total degree $d$, there are also precise bounds 
on the coefficients $c\_{\alpha}$ with $|\alpha|=d$ or $d-1$ (for the case of the ball or the cube).


In that connection, two interesting papers are



> 
> M.I. Ganzburg, A Markov-type inequality for multivariate polynomials
>  on a convex body. J. Comput. Anal. Appl. 4 (2002), no. 3, 265-268
> 
> 
> H-J. Rack, On V. A. Markov's and G. Szegő's inequality for the
>  coefficients of polynomials in one and several variables. East J.
>  Approx. 14 (2008), no. 3, 319-352,
> 
> 
> 


and the bibliography therein.


