Analytic extension of functions in Hardy spaces

Question

This is a problem I came across in a direct scattering problem. I have a function $a(s)$ that is of the form$$ a(s)=\int_0^{\infty}e^{is\xi}A(\xi)d\xi $$ where $A(\xi)\in L^1\cap L^2$. Then is it possible to extend this function to a bounded analytic function in $\mathbb{C}^{+}$? Why or why not? Can someone give me a proof? I know that if taking $s$ to be a complex number $z\in\mathbb{C}^{+}$, then the new function $a(z)$ lies in the Hardy space $H^2(\mathbb{C}^{+})$. I tried to evaluate the curve integral$$ \oint_{\gamma}a(z)dz $$so that I can use Morera theorem. But how to calculate this? Can someone help me?

Answer 1

To see that $a(s)$ is bounded, note this

$$ |a(s)|=\Bigg|\int_0^{\infty}e^{is\xi}A(\xi)\,d\xi\, \Bigg|\leq \int_{0}^{\infty}|A(\xi)|d\xi < \infty, $$

since $A(\xi) \in L_1(0,\infty)$.