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 $m$. (That is, 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 $m$.)
Is it 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 $\mathbb{R}^d$?
That is, does there exist a compact set $K$ in $\mathbb{R}^d$ together with 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.
Edit: Do you know if $\kappa$ can be chosen independent of $m$, or at least such that the sequence $(\kappa(m,d,K))_{m\geq 0}$ is bounded (for $K$ and $d$ fixed)?
Best Answer
Yes, such a $\kappa$ exists for every compact set $K$ with non-empty interior. Here is an abstract linear-algebra argument.
Let $V$ be the real vector space spanned by the multi-indices $\alpha$ with length at most $m$. We have a linear map $A\colon V \to \mathbb R^K$, sending $(c_\alpha)$ to the function $x\to \sum_\alpha c_\alpha x^\alpha$. Since different polynomials cannot agree on a set with non-empty interior, this map is injective. There is then a finite subset $F\subset K$ such that the composition $V\xrightarrow A\mathbb R^K\to\mathbb R^F$ is a linear isomorphism (for example, pick the elements of $F$ one by one, making sure that the rank of the composition increases by 1 each time).
Now $\kappa$ exists by the fact that all linear isomorphisms between finite-dimensional vector spaces are bicontinuous.
If you want an explicit value for $\kappa$, you can use discrete derivatives (see https://en.wikipedia.org/wiki/Finite_difference) to write $c_\alpha$ as a linear combination of some values of $P$.