Analytic functions characterized by their means on an arc

Abstract:

It is known that a function f, holomorphic in the open unit disc U with ${C^{1 + \varepsilon }}$ boundary data for some $\varepsilon > 0$, is uniquely determined by its arithmetic means over equally spaced points on $\partial U$. By using different techniques, we weaken the hypothesis ${C^{1 + \varepsilon }}(\partial U)$ to functions with ${L^p}$ derivatives, $1 < p \leq \infty$. We also prove that a function is determined by its averages over an arc K if f is holomorphic in a neighborhood of $\bar U$, and that this result is false for some functions f in $A \cap {C^\infty }(\bar U)$. On the other hand, we can capture a $A \cap {C^2}(\bar U)$ function from its means and shifted means on K.