Square integrable functions on the circle

Question

Consider the Hilbert Space $L^2(S^1)$, the space of all square integrable functions on the unit circle. My question is whether all functions in this space can be written as a sum $\sum_{n=0}^\infty a_nz^n$ for $a_n\in \mathbb{C}$.

I think this has something to do with Fourier Series, I seem to have forgotten such things. It would be very nice if someone indicated where can I read about this. Thank you.

Answer 1

Regardless of the meaning of convergence, the answer is negative. The functions $\{z^n : n=0,1,2,\dots\}$ do not form a basis of $L^2(S^1)$ in any sense. Indeed, the function $\bar z$ (or $1/z$, which is the same thing on the circle) is orthogonal to all of them: $$ \int_{S^1} \bar z z^n = 0,\quad n=0,1,\dots $$

But one can write $f(z) = \sum_{n=-\infty}^\infty a_nz^n$, using both positive and negative powers of $z$. Convergence holds

• in the sense of $L^2$ norm, that is $\left\| f - \sum_{|n|\le N} a_nz^n \right\|_2\to 0$ as $N\to\infty$;
• pointwise at almost every point of $S^1$ (Carleson's theorem)

The Wikipedia article Convergence of Fourier series is a good place to start.