If I start with the set of powers of $x$ excluding $x^2$, that is $\{1, x, x^3, x^4 \dots\}$ and perform Gram-Schmidt orthonormalization in the interval $-1 \le x \le 1$, will the resulting basis still produce convergence in the mean for a well-behaved function?
"Obviously" I cannot match term-for-term any polynomial expression that involves $x^2$, since it is not in my basis, but given the set of basis functions $Q_n(x)$ from the above process and any reasonable function $f(x)$, does $$ \lim_{n\to\infty} \left|\sum_n\, (Q_n,f)\,\, Q_n(x) - \, f(x) \right|^2 =0 $$ that is, we can approximate $f(x)$ in the mean?
I had thought that the answer was "no", but numerical evidence is against me, and I can't find a function that is "unexpandable" in this basis.