# Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

## TL;DR

This paper investigates the ill-posed nature of analytic continuation problems, demonstrating that solutions exhibit power law precision decay, and introduces a Hilbert space-based method to determine these decay exponents explicitly.

## Contribution

The paper develops a Hilbert space framework to explicitly compute power law exponents in analytic continuation, including solutions in special geometries like the annulus and upper half-plane.

## Key findings

- Power law decay of solution accuracy away from data source
- Explicit formulas for exponents in circular geometries
- Method aligns with prior results in special cases

## Abstract

The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the "worst case" function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.