1
$\begingroup$

For any integer $n\geq 1$ let $S^n$ denote the unit sphere in $\mathbb{R}^{n+1}$. There is a natural action of $\Sigma_{n+1}$ (the symmetric group on $n+1$ letters) on $S^n$.

Is there a family of real-analytic functions $f_n:S^n\to \mathbb{R}$ such that

  • for any given $n\geq 1$ the function $f_n$ is $\Sigma_{n+1}$-invariant
  • $f_n(a_1, \dots, a_{n+1})=f_{n+1}(a_1, \dots, a_{n+1}, 0)$
  • $\bigcup_{n\geq 1} f_n(S^n)\subset \mathbb{R}$ is a bounded set
  • $f_1(a_1, a_2)=a_1+a_2$?

For example $f_n(a_1, \dots, a_{n+1})=\sqrt{\frac{2}{n+1}}\sum_{i=1}^{n+1}a_i$ satisfies all but one condition.

$\endgroup$

1 Answer 1

3
$\begingroup$

Yes, there even exist symmetric polynomials with such properties.

Let $\sqrt{2}=c_1<c_2<c_3<\ldots$ be a bounded sequence. Our aim is to construct inductively symmetric polynomials $f_1,\ldots,f_n$, on the spheres $S^1,\ldots,S^n$ respectively such that $f_1(a_1,a_2)=a_1+a_2$, $\max |f_i|\leqslant c_i$ and $f_n(a_1,\ldots,a_{n+1})=f_{n+1}(a_1,\ldots,a_{n+1},0)$.

Assume that $f_n$ is constructed. Denote $\Delta=\{(x_1,\ldots,x_{n+2})\in S^{n+2}:\prod x_i=0\}$.

First of all, construct some symmetric polynomial $G(x_1,\ldots,x_{n+2})$ such that $G(x_1,\ldots,x_{n+1},0)=f_n(x_1,\ldots,x_{n+1})$ (this is easy to do by symmetrizing all monomials of $f_n$.)

Next, construct a symmetric continuous function $h(x_1,\ldots,h_{n+2})$ which coincides with $G$ on a small neighborhood $V$ of the set $\Delta$ and satisfies $\max |h|\leqslant (c_n+c_{n+1})/2$. There are many ways to do it, for example by solving Dirichlet Laplacian problem on $S^{n+1}\setminus V$, we only need to choose $V$ so that $|G|<(c_n+c_{n+1})/2$ on $V$.

Finally denote $p=\frac{G-h}{x_1\ldots x_{n+2}}$. This is a continuous symmetric function on $S^{n+2}$ (vanishing on $V$), it may be approximated by a symmetric polynomial $q$ uniformly on $S^{n+2}$ with accuracy $(c_{n+1}-c_n)/2$. Put $f_{n+1}=G-q\cdot \prod x_i$. It is a symmetric polynomial coinciding with $G$ on $\Delta$ and satisfying $$ |f_{n+1}(x_1,\ldots,x_{n+2})|=|G-q\prod x_i|\leqslant |G-p\prod x_i|+|p-q|\leqslant \max |h|+\max |p-q|\leqslant c_{n+1}. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.