Representation of $L^2$ and $H^2$ spaces by sequences

Question

The Hardy space $H^2$ can be viewed as a closed vector subspace of the complex $L^2$ space on the unit circle. Can you please explain why the $L^2$ space can be represented by bi-infinite sequences and $H^2$ can be represented by infinite sequences $f(z) = \sum_{n=0}^\infty a_n x^n, |z|<1$ ?

Answer 1

The space $L^2(\mathbb{T})$ of functions on the unit circle $\mathbb{T}$ has inner product $$ \langle f,g\rangle = \int_{\mathbb{T}}f(z)\overline{g(z)}d\sigma(z) $$ where $\sigma$ is the Lebesgue measure on $\mathbb{T}$, normalized so that $\sigma(\mathbb{T})=1$. This space has an orthonormal basis $\{ z^n\}_{n=-\infty}^{\infty}$. The functions in $H^2(\mathbb{T})$ consist of those with holomorphic extensions to the unit disk $\mathbb{D}$. $H^2(\mathbb{D})$ has an orthonormal basis $\{ z^n \}_{n=0}^{\infty}$.

Every $f\in L^2(\mathbb{T})$ may be written as the $L^2$ convergent sum $$ f(z) = \sum_{n=-\infty}^{\infty}a_n z^n,\;\;\; z\in\mathbb{T}. $$ The functions in $H^2(\mathbb{T})$ are those for which $\langle f,z^n\rangle=0$ for $n < 0$. The functions in $H^2(\mathbb{T})$ have natural extensions to unit disk $\mathbb{D}$ as a power series $f(z) = \sum_{n=0}^{\infty}a_n z^n$, and it not difficult to show that $f_{r}(z)=f(rz)$ converges to $f$ in $L^2(\mathbb{D})$ and $r\uparrow 1$.