Polynomial that grows faster than any other polynomial outside $[−1,1]^n$

Question

Consider this statement: "Chebyshev polynomials increase in magnitude more quickly outside the range $[−1,1]$ than any other polynomial that is restricted to have magnitude no greater than one inside the range $[−1,1]$."

In the multivariate case, is there a degree $d$ polynomial that lies in $[-1,1]$ in the interval $[-1,1]^n$ and grows faster than any other polynomial of degree $d$ outside the $[-1,1]^n$? If so, what is this polynomial? It is natural to conjecture that $q(x_1,..,,x_n) = T_{d/n}(x_1)T_{d/n}(x_2)..T_{d/n}(x_n)$ (assuming $n|d$) is this polynomial ($T_i(x)$ is the degree $i$ Chebyshev polynomial).

For the univariate case, a proof can be found here.

I have tried to generalize this proof to the bivariate case, but the fact that bivariate polynomials can have infinitely many roots (while univariate polynomials can have only at most degree-many roots) seems to be a barrier in adapting the proof.

Answer 1

Consider the 2D case, and the line $x=x_0$, for $x_0 \in (-1,1)$. Along that line, the fastest growing polynomial is a Chebyshev polynomial.

A Chebyshev polynomial in $y$ multiplied by $f(x)=1$ will make a "multivariate" polynomial that is in $[-1,1]$ in the unit box, and grows faster than any polynomial on $[-1,1] \times ((-\infty,-1) \cup(1,\infty))$.

However, this polynomial grows much more poorly outside of that region.

I think you need to relax your definition of growth. No single multivariate polynomial will grow most quickly everywhere.

Answer 2

The product of Chebyshev polynomials does not seem to be the fastest growing polynomials outside the $L_{\infty}$ unit ball, at least for $n=2$. It looks like that for all even $d$, $x^d/3+y^d/3$ is greater in absolute value than $T_{d/2}(x)T_{d/2}(y)$ in some region outside the $L_{\infty}$ unit ball.

See the Desmos plots for $d=$ $2$, $4$ and $6$.

These examples in fact shows that, for a fixed $p \geq 1$, $n$, and $d$, the product of Chebyshev polynomials (with individual degrees $d/n$) need not grow faster all other degree $d$ polynomials outside the unit ball in $L_p$ norm. Hence, some of the other natural ways to generalize the one-dimensional case to higher dimensions is ruled out.

Whether there exist degree $d$ polynomials (bounded in the $L_{\infty}$ unit ball) that grows faster than all other degree $d$ polynomials (that are bounded in the unit ball) outside the unit ball is still not clear.