3
$\begingroup$

Given a function $f(z)$, $z=x+iy, x,y\in \mathbb R$, which belongs to $\mathbb H^{2}(\mathbb C^{+})$, where $\mathbb C^{+}$ is the upper half plane Im(z)>0 and $f(a_{n})=0$, for all $n\in \mathbb Z$, where a_{n}'s are all real numbers. What can we say about this function $f$? should it be the zero function!? or there is something else that we can say about?

Edit: $f$ is continuous on the real line, and has singularities in the lower half plane, and the sequence $\{a_{n}\}$ has no accumulation point.

  • 0
    This theorem might be helpful. Theorem (Fatou):Let $K$ be a closed set of Lebesgue measure zero on the unit circle. Then there exist a function which is continuous on closed disk $\bar{\mathbb{D}}$ and holomorphic on open unit disk $\mathbb{D}$ and vanishes precisely on $K$.2015-05-09

0 Answers 0