Consider the interval $[0,1]$ and let $\mu_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu_1(t)<\mu_2(t)<\ldots<\mu_n(t)$ for all $t \in [0,1]$.

Consider for each $k \geq 1$, the functions $$f_k(t)=\sum_{j=1}^n (\mu_j(t))^k$$

Is it true that any continuous function on $[0,1]$ can be approximated uniformly by linear combinations of the functions $f_1,f_2,f_3,\ldots$?

The case $n=1$ clearly holds due to the Stone-Weierstrass theorem but I can't see the general case. Thanks for your help.

Consider the interval $[0,1]$ and let $\mu_k(t)$ with $k=1,\ldots,n$ be continuous functions such that they are all strictly increasing on the interval $[0,1]$ and such that $\mu_1(t)<\mu_2(t)<\ldots<\mu_n(t)$ for all $t \in [0,1]$.

Consider for each $k \geq 0$, the functions $$f_k(t)=\sum_{j=1}^n (\mu_j(t))^k$$

Is it true that any continuous function on $[0,1]$ can be approximated uniformly by linear combinations of the functions $f_0,f_1,f_2,f_3,\ldots$?

The case $n=1$ clearly holds due to the Stone-Weierstrass theorem but I can't see the general case. Thanks for your help.