| Abstract: |
| Analytic functions in a domain $\Omega$ are uniquely determined by their values on any curve $\Gamma \subset \Omega$. We provide sharp quantitative version of this statement. Namely, let $f$ be of order $\epsilon$ on $\Gamma$ w.r.t. some Hilbert space norm (e.g. $L^2$) and of order 1 on $\Omega$ w.r.t. some other Hilbert space norm. How large can $f$ be at a point $z$ away from the curve? We give a sharp upper bound on $|f(z)|$ in terms of a solution of an integral equation and demonstrate that the bound behaves like a power law: $\epsilon^{\gamma(z)}$. In special geometries, such as the upper half-plane, annulus or ellipse the integral equation can be solved explicitly, giving exact formulas for the exponent $\gamma(z)$. |
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