Family of symmetric functions on unit spheres with a growth condition

Question

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.

Answer 1

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 $$ |f_{n+1}(x_1,\ldots,x_{n+2})|=|G-q\prod x_i|\leqslant |G-p\prod x_i|+|p-q|\leqslant \max |h|+\max |p-q|\leqslant c_{n+1}. $$