How to show that $(1, x, x^2, ..., x^n, \log x , x\log x, ..., x^m\log x)$ form a Chebyshev system?

Question

How to show that $(1, x, x^2, ..., x^n, \log x , x\log x, ..., x^m\log x)$ form a Chebyshev system of order $n+m+2$ on $(0,1)$ whenever $n\geq m$?

Thanks!

Answer 1

We know that a linear combination of $\{1,\ldots x^n\}$ can have at most $n$ roots. Same is true for a linear combination of $\{\tfrac1{x^k},\ldots \tfrac{x^n}{x^k}\}$ for $x\neq0$, $k\geq0$.

Assume that a linear combination of $\{1,\ldots x^n,\log x\}$ could have $r>n+1$ roots. Then by Rolle's theorem its derivative, itself a linear combination of $\{\tfrac1{x^1},\ldots \tfrac{x^n}{x^1}\}$, must have $r'>n$ roots. Contradiction.

And so on, either by induction or by differentiating $m+1$ times with the condition $m\leq n$, we will conclude that a linear combination of $\{1,\ldots x^n,\log x\ldots x^m\log x\}$ cannot have more than $n+1+m$ roots.