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    Quantification of stability of analytic continuation with applications to electromagnetic theory

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    Hovsepyan_temple_0225E_14484.pdf
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    Genre
    Thesis/Dissertation
    Date
    2021
    Author
    Hovsepyan, Narek
    Advisor
    Grabovsky, Yury
    Committee member
    Berhanu, Shiferaw
    Gutiérrez, Cristian E., 1950-
    Harutyunyan, Davit
    Subject
    Mathematics
    Applied mathematics
    Analytic continuation
    Extrapolation
    Herglotz functions
    Optimal error estimates
    Power law
    Permanent link to this record
    http://hdl.handle.net/20.500.12613/6808
    
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    DOI
    http://dx.doi.org/10.34944/dspace/6790
    Abstract
    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.
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