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 nonnegative 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 nonnegative 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 GramSchmidt 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 welldefined. It's just the usual choice. 


I'll answer this question for the interval [1,1], but you should be able to extrapolate/specialize it to your case. Let me define the inner product $\langle f,g\rangle=\int_{1}^1 f(x)g(x) dx$ and normalized Legendre polynomials $P_m(x) = \sum_{n=0}^m p_{mn}x^n$ such that $\langle P_m,P_{m'}\rangle=\delta_{mm'}$. Note that $p_{mn}=0$ for $m>n$, so that the matrix $P$ with coefficients $p_{mn}$ is upper triangular. Let $Q$ be the inverse matrix, with components $q_{nm}$, that is, $x^n = \sum_{m=0}^n q_{nm}P_m(x)$. Clearly, $Q$ is also upper triangular. The triangularity matters because $Q$ can be found by explicit calculation, even though both $P$ and $Q$ are infinite dimensional. It easily follows that $\langle x^n,P_m\rangle = q_{nm}$. It immediately follows that $\langle x^n,\sum_m p_{mn'} P_m\rangle = \sum_m q_{nm}p_{mn'} = \delta_{nn'}$. In other words, the functions you want are $u_{n}(x) = \sum_{m=0}^n p_{mn} P_m(x)$. 


Derivatives of the Dirac distribution (appropriately normalized). 

