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For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (I.e., the above sum ranges over all multiindices $\alpha=(i_1, \ldots, i_d)\in\mathbb{N}_0^{\times d}$ of length $|\alpha|\equiv i_1+\ldots + i_d$ less than $m$.)

I was wondering if it is somehow possible to estimate the maximum coefficient $\|c_\alpha\|_\infty := \max_{|\alpha|\leq m}|c_\alpha|$ of $P$ against its uniform norm $\|P\|_{\infty;K}:= \sup_{x\in K}|P(x)|$, for $K$ some compact set in $\mathbb{R}^d$ (ideally something like the closed unit ball, but whatever compact set works).

That is, does there exist a constant $\kappa \equiv \kappa(m,d,K)>0$ such that

$$\tag{1}\|c_\alpha\|_\infty \ \leq \ \kappa\cdot \|P\|_{\infty; K} \qquad \text{ for each } \ P \ \text{ as above}?$$

Any references are welcome.

fsp-b
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    If you look at Tchebycheff polynomials with m =1 one gets estimates of the order $2^d$ so nothing better can be expected. – Salcio Dec 16 '21 at 17:19

2 Answers2

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One thing you can do is pick out the coefficient $c_\alpha$ for any $\alpha$ using some variant of the Cauchy integral formula. By $d$ applications of Cauchy we have $$\frac{1}{(2\pi i)^d} \int_{(S^1)^d} P(z) z^{-\alpha-1} dz = c_\alpha.$$ It follows that $$|c_\alpha| \leq \max_{z \in (S^1)^d} |P(z)|.$$

I doubt you can get away with a compact subset of $\mathbb{R}^d$, even if $d=1$.

Sean Eberhard
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    But estimating in this way you are going into complex variables. The question was for real parameters. Even in the one dimensional case one can easily come with examples of polynomials which are bounded by 1 on $[-1,1]$ and explode on circle. – Salcio May 10 '23 at 21:05
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I've recently been looking into this problem. Here's what I've came up with to show the existence of such a $\kappa$. Let me know if there are any issues.

Define the following polynomial space $$ \mathcal{P}_{m,d,K} := \left\{ P: P(x) = \sum_{\alpha:|\alpha|\leq m} c_\alpha x^\alpha, \quad x\in K, \;c_\alpha \in \mathbb{R} \; \forall \alpha \right\} . $$ It can be shown that $\mathcal{P}_{m,d,K}$ is a vector space. Further, since every $P\in\mathcal{P}_{m,d,K}$ can be represented as a linear combination of finitely many basis functions ($x^\alpha$), then $\mathcal{P}_{m,d,K}$ is a finite dimensional vector space.

Then we have the following two polynomial norms, which are both valid norms on $\mathcal{P}_{m,d,K}$: $$ \lVert P \rVert_{\infty;\,K} := \sup_{x\in K}|P(x)| = \max_{x\in K}|P(x)|, $$ and $$ \lVert P \rVert_{\infty;\,c_\alpha} := \max_{\alpha:|\alpha|\leq m}|c_\alpha| = \lVert c_\alpha \rVert_\infty. $$ It is a well known result that all norms on finite dimensional vector spaces are equivalent (Link1, Link2). Therefore, from the definition of equivalent norms, there exists a $\kappa>0$ such that
$$ \lVert P \rVert_{\infty;\,c_\alpha} \leq \kappa\lVert P \rVert_{\infty;\,K} . $$

Chad Brown
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