Analytic functions in a domain Ω are uniquely determined by their values on any curve Γ ⊂ Ω. We provide sharp quantitative version of this statement. Namely, let f be of order E on Γ relative to its global size in Ω (measured in some 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 a linear integral equation of Fredholm type and demonstrate that the bound behaves like a power law: E^γ(z). In special geometries, such as the upper halfplane, annulus or ellipse the integral equation can be solved explicitly, giving exact formulas for the optimal exponent γ(z). Our methods can be applied to
non-Hilbertian settings as well. Further, we apply the developed theory to study the degree of reliability of extrapolation of the complex electromagentic permittivity function based on its analyticity properties. Given two analytic functions, representing extrapolants of the same experimental data, we quantify how much they can differ at an extrapolation point outside of the experimentally accessible frequency band.
| Identifer | oai:union.ndltd.org:TEMPLE/oai:scholarshare.temple.edu:20.500.12613/6808 |
| Date | January 2021 |
| Creators | Hovsepyan, Narek |
| Contributors | Grabovsky, Yury, Berhanu, Shiferaw, Gutiérrez, Cristian E., 1950-, Harutyunyan, Davit |
| Publisher | Temple University. Libraries |
| Source Sets | Temple University |
| Language | English |
| Detected Language | English |
| Type | Thesis/Dissertation, Text |
| Format | 111 pages |
| Rights | IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available., http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | http://dx.doi.org/10.34944/dspace/6790, Theses and Dissertations |
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