I would like to ask if there are any set of functions $u\_n(x)$ which is orthogonal to $x^n$?
i.e.:


$\int\_0^1 x^n u\_m(x) dx = \delta\_{n,m}$


Edit: For clarification, this question asked for all non-negative integer m and n.



Functions orthogonal to x^n
If $f \in L^2([0, 1])$ and $\int\_0^1 x^n f(x)\, dx=0$ for all $n\ge N$ where $N$ is a nonnegative integer then $f$ is zero almost everywhere. To see this note that $x\mapsto x^N f(x)$ is an $L^2$ function orthogonal to all polynomials, and the polynomials are dense in $L^2([0,1])$. So the answer to your question is "no" for $L^2$ functions.


