Hello!

Could you please help with the following question:

Let $\displaystyle A^2 (\mathbb{D} ) $ be the Bergman space of all holomorphic functions on the unit disc $\displaystyle \mathbb{D} $ which also belong to $\displaystyle L^2 ( \mathbb{D}) $. Let $\displaystyle f \in A^2(\mathbb{D}), \ 0<s<1 $, and $\displaystyle |z| < s$. Cauchy's Integral Formula gives:

$\displaystyle

rf(z) = \frac{1}{2 \pi} \int_{0}^{2 \pi} f(z + re^{i \theta}) r d \theta \ (0<r<1-s).

$

By integrating this formula for $\displaystyle 0<r<1-s $, we are supposed to show that

$\displaystyle f(z) = \frac{1}{\pi (1-s)^2} \langle f, \chi_{\mathbb{D}(z,1-s)} \rangle_{L^2 (\mathbb{D})}

$,

where $\displaystyle \mathbb{D} (z, 1-s) $ is the disc of radius $\displaystyle 1-s$ with centre $\displaystyle z$.

From this, we should deduce that $\displaystyle |f(z)| \leq \frac{\| f \| _L^2 (\mathbb{D})}{(1-s)\surd{ \pi} }$

From this I know how to deduce that $\displaystyle A^2 (\mathbb{D}) $ is closed in $\displaystyle L^2 (\mathbb{D})$; it's just the above calculations which confuse me.