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.
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.
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.
The answer is no, and the main reason is that $\{x^n\}_{n = 0}^{\infty}$ form a total set in $L^2([0,1])$ so the set of their finite linear combinations is dense. But $\int x^n dx > 0$ for $n \geq 0$.
I believe, the best one can do is apply Gram--Schmidt to $x^n$ and obtain a sequence of polynomials $p_n$ (the orthogonal polynomials) of degree $n$ such that $$ p_n \perp x^m,\quad m > n. $$ Here $f \perp g$ means $\int f(x) g(x) dx= 0$.
However, the notion of "best" here is not well-defined. It's just the usual choice.
Derivatives of the Dirac distribution (appropriately normalized).