Orthogonal complement to a span of monomials

Question

Whereas it is clear that in $L^2[-1,1]$ (as a Hilbert space on $\mathbb{C}$) the span of all monomials $1,x,x^2,x^3,\dots$ is dense (Stone-Weierstrass), and hence its orthogonal complement is $\{0\}$, it is not clear to me how the conclusion changes if one only considers the monomials $x^k,x^{k+1},x^{k+2},\dots$, i.e., starting from some non-zero order $k$.

On the one hand, again by Stone-Weierstrass, it seems correct to me to claim that the subspaces $$ W_k:=\mathrm{span}\{1,x,\dots,x^{k-1}\}\qquad \textrm{and}\qquad V_k:=\mathrm{span}\{x^k,x^{k+1},x^{k+2},\dots\} $$ are linearly independent and that the direct sum $W_k\dotplus V_k$ (namely the polynomials) is dense in $L^2[-1,1]$. I'd be then led to deduce that $$\dim V_k^\perp=\dim W_k=k.$$

On the other hand, it seems also correct to me to say that $$ V_k=\{x^k p\,|\,p\,\textrm{is a polynomial in $x$}\}\,, $$ and therefore, if $u$ is a generic function in $V_k^\perp$, then $$ 0=\int_{-1}^1\overline{u}\,x^k p\,dx\qquad\forall\,\textrm{polynomial } p\,. $$ But then the conclusion seems to be that $x^k u=0$, whence $u=0$ in $L^2[-1,1]$.

Where am I wrong? What is the space $V_k^\perp$ for generic positive integer $k$?

Answer 1

It is the latter conclusion that is true. That is, the dimension of the perpendicular $V_k^\perp$ is $0$, not $k$. Recalling the definition,

$$ V_k^\perp = \{ q \in L^2 : \langle q,p \rangle = 0\ \forall p \in V_k\}$$ Then $1\notin V_k^\perp $ since $V_k$ contains $x^{2N}$ for some $N$, so $$ \langle 1, x^{2N} \rangle = \|x^{2N}\|_{L^1} > 0$$

(this is nothing but a special case of your latter argument)

This makes some sense to me; suppose we wanted to approximate some function $f$ in $L^2$ using only polynomials in $V_k$. Without loss, $f$ is continuous, but we can also further change $f$ into a function $\tilde f$ that is $0$ on a small neighbourhood of $0$ by incurring some small cost $\|f-\tilde f\|_2 < \epsilon$. After doing that, we can approximate $\tilde f/x^k\in C^0$ uniformly with polynomials, so that $$ \|f -px^k\|_2 \le \|f-\tilde f\|_2 + \|\tilde f/x^k -p\|_\infty\|x^k\|_2 \le \epsilon + \|\tilde f/x^k -p\|_\infty\|x^k\|_2 $$ improving the choice of $p$ at this point makes this arbitrarily small, hence the result.

Answer 2

That is the answer I gave to myself.

$V_k$ is a (complex) sub-algebra of $C[-1,1]$ (with obviously defined sum and product), which separates points and such that all its elements vanish at $x=0$. Therefore, by the Stone-Weierstrass theorem, the closure of $V_k$ in the uniform norm is $$ \overline{V_k}^{\|\,\|_{\infty}}\;=\;\{h\in C[-1,1]\,|\,h(0)=0\}\,. $$ The latter space contains $C^\infty_0(-1,1)$, therefore by standard approximation facts it is dense in $L^2$ (in the $L^2$-norm!). Since on the compact $[-1,1]$ the $L^\infty$-norm controls the $L^2$-norm, $V_k$ is also dense in $L^2[0,1]$ (in the $L^2$-norm!)

Answer 3

If $\int_{-1}^{1}f(x)x^kdx = 0$ for all $k \ge n$ for some $n > 0$, then $\int_{-1}^{1}(f(x)x^n)x^{k}dx=0$ for all $k \ge 0$, from which it follows that $f(x)x^n=0$ for all $x$. So $f(x)=0$ for all $x\in [-1,1]\setminus\{0\}$. Because $f$ is continuous, then $f(x)=0$ for all $x\in [-1,1]$.