Let $p:=\sum_{n=0}^\infty p_n$ be a polynomial given by its terminating decomposition by means of homogeneous polynomials. For fixed $x\in \mathbb{R}^d $ and an none negative integer $n$, can we find a compact $K_{x,n}\subset \mathbb{R}^d$ and a constant $C_{x,n}$ such that $|p_ n(x)|\le C_{x,n} \sup_{z\in K_{x,n}}|p(z)|$ ?

Thanks in advance.

  • $\begingroup$ @J.C. Otten: I think you overlooked that the left side of the desired inequality involves $p_n$ while the right side involves $p$. $\endgroup$ – Andreas Blass May 18 '11 at 17:45
  • $\begingroup$ Following your response, if for fixed x we have p(x)=0 then p_n(x)=0. !!!!! Sorry you have not seen $p$ and $p_n$ are both involved. $\endgroup$ – mostafa May 18 '11 at 17:50
  • $\begingroup$ I am confused about quantifiers here. Are $K_{x,n}$ and $C_{x,n}$ allowed to depend on $p$? $\endgroup$ – Yemon Choi May 18 '11 at 17:57
  • $\begingroup$ Well, if everything is dependent of $p$ this is trivial. For a closed ball $B$ of radius $r$ around $x$, the maximum of $p$ over $B$ will approach infinity as the $r$ grows large. In particular, it will be larger than $|p_n(x)|$ for $r$ large enough.. $\endgroup$ – J.C. Ottem May 18 '11 at 18:01
  • $\begingroup$ JCO: that's what I thought, but I wanted the OP to clarify. $\endgroup$ – Yemon Choi May 18 '11 at 18:11

Here's a counterexample with $d=1$. Since $d=1$, we have $p_n(x)=a_nx^n$, so $p(x)=\sum_{n=0}^\infty a_nx^n$. Since you call this a polynomial, we'll also assume that $a_n=0$ for sufficiently large $n$. Let's take a special case of what you'd like to be true: $$ |p_2(1)|\leq C\sup_{x\in K}|p(x)| $$ for some compact $K\subseteq\mathbb R$ and some $C<\infty$. In other words: $$ |a_2|\leq C\sup_{|x|\leq N}|p(x)| $$ Suppose this is true for all polynomials $p(x)$, and apply it to a partial taylor expansion of $f_A(x)=e^{-Ax^2}$, $A>0$. Then $a_2=\frac 12f_A''(0)=-A$, so your inequality would imply that $|A|\leq C\sup_{|x|\leq N}|p(x)|$. However the power series of $f_A(x)$ has infinite radius of convergence, so no matter what $N$ is, we can choose a very high degree taylor approximation which equals $f_A$ within an error of $\frac 1{10}$ on $[-N,N]$. Thus the right hand side will be bounded by, say $2C$. On the other hand, we are free to choose $A$ as large as we want, so this is a contradiction.

There are two ways I see to fix the desired inequality. First, you could assume bound the degree of $p(x)$. Then the space of such polynomials is finite dimensional, and in this case any $K$ with nonempty interior works, since then $\sup_{z\in K}|p(z)|$ is a norm on this space of such polynomials. (incidentally, this is essentially related to Problem A5 on the 1999 Putnam, see http://amc.maa.org/a-activities/a7-problems/putnamindex.shtml).

Second, you could instead consider polynomials on the complex numbers, and allow supremum on the right hand side to be over compact $K\subseteq\mathbb C^n$. Then the Cauchy Integral Formula (http://en.wikipedia.org/wiki/Cauchy_integral_formula) would make your statement true (at least for $d=1$, and probably all $d$ as well).

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