Optimal Error Estimates for Analytic Continuation in the Upper Half-Plane

Analytic functions in the Hardy class H-2 over the upper half-plane & x210d;(+) are uniquely determined by their values on any curve Gamma lying in the interior or on the boundary of & x210d;(+). The goal of this paper is to provide a sharp quantitative version of this statement. We answer the following question: Given f of a unit H-2-norm that is small on Gamma (say, its L-2-norm is of order epsilon), how large can f be at a point z away from the curve? When Gamma subset of partial differential & x210d;(+), we give a sharp upper bound on divide f(z) divide of the form epsilon(gamma), with an explicit exponent gamma = gamma(z) is an element of (0, 1) and explicit maximizer function attaining the upper bound. When Gamma subset of & x210d;(+) we give an implicit sharp upper bound in terms of a solution of an integral equation on Gamma. We conjecture and give evidence that this bound also behaves like epsilon(gamma) for some gamma = gamma(z) is an element of (0, 1). These results can also be transplanted to other domains conformally equivalent to the upper half-plane. (c) 2020 Wiley Periodicals LLC

Automated summary

Analytic functions in the Hardy class H-2 over the upper half-plane are uniquely determined by their values on any curve Γ lying in the interior or on the boundary of the upper half-plane. This paper aims to provide a sharp quantitative version of this statement, giving explicit upper bounds in certain cases and implicit upper bounds in others. The results can be extended to other domains conformally equivalent to the upper half-plane.