scholarly journals OVERVIEW OF VNC PUBLICATIONS 

2018 ◽  
Author(s):  
Г.П. Селиверстова
Keyword(s):  
Fourier Transform ◽  
Dirichlet Problem ◽  
Half Plane ◽  
Boundary Function ◽  
Optimal Recovery ◽  
Recovery Method ◽  
Finite Segment ◽  
Plane Parallel ◽  
Straight Line ◽  
Upper Half Plane

Работа посвящена задаче наилучшего восстановления реше- ния задачи Дирихле в пространстве квадратично суммируемых функций на прямой в верхней полуплоскости, параллельной оси абсцисс, по следующей информации о граничной функции: гра- ничная функция принадлежит некоторому соболевскому про- странству функций, а ее преобразование Фурье известно при- ближенное на конечном отрезке, симметричном относительно нуля. Построен оптимальный метод восстановления и найдено точное значение погрешности оптимального восстановления. The work is devoted to the problem of the best recovery solution- the Dirichlet problem in the space of quadratically summable functions on a straight line in the upper half plane parallel to the axis abscissa, on the following information on the boundary function: gra- personal function belongs to some Sobolev Pro- the wandering of functions, and its Fourier transform is known in- near on a finite segment, symmetric with respect to zero's. The optimal recovery method was constructed and found the exact error value of the optimal recovery.

scholarly journals Scattering on periodic metric graphs

2020 ◽  
Vol 32 (08) ◽  
pp. 2050024
Author(s):  
Evgeny Korotyaev ◽  
Natalia Saburova
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Fredholm Determinant ◽  
Unit Interval ◽  
Discrete Laplacian ◽  
Metric Graphs ◽  
Wave Operators ◽  
S Matrix ◽  
Upper Half Plane ◽  
Uniformly Bounded

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are uniformly bounded. We apply these results to the periodic metric Laplacian perturbed by real integrable potentials. We prove the following: (a) the wave operators exist and are complete, (b) the standard Fredholm determinant is well-defined and is analytic in the upper half-plane without any modification for any dimension, (c) the determinant and the corresponding S-matrix satisfy the Birman–Krein identity.

A Dirichlet Problem for the Inhomogeneous Polyharmonic Equation in the Upper Half Plane

2007 ◽  
Vol 14 (1) ◽  
pp. 33-52
Author(s):  
Heinrich Begehr ◽  
Evgenija Gaertner
Keyword(s):  
Dirichlet Problem ◽  
Green Function ◽  
Differential Operator ◽  
Integral Representation ◽  
Half Plane ◽  
Representation Formula ◽  
Higher Order ◽  
Polyharmonic Equation ◽  
Integral Representation Formula ◽  
Upper Half Plane

Abstract On the basis of a higher order integral representation formula related to the polyharmonic differential operator and obtained through a certain polyharmonic Green function, a Dirichlet problem is explicitly solved in the upper half plane.

A note on holomorphic functions and the Fourier-Laplace transform

2017 ◽  
Vol 120 (2) ◽  
pp. 225 ◽  
Author(s):  
Marcus Carlsson ◽  
Jens Wittsten
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Half Plane ◽  
Classical Problem ◽  
Holomorphic Functions ◽  
Function Spaces ◽  
Classical Function ◽  
Upper Half Plane ◽  
The Fourier Transform ◽  
Half Axis

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

scholarly journals Quasiconformality of harmonic mappings between Jordan domains

Filomat ◽  
2012 ◽  
Vol 26 (3) ◽  
pp. 479-510 ◽  
Author(s):  
Miodrag Mateljevic
Keyword(s):  
Half Plane ◽  
Unit Disc ◽  
Necessary Conditions ◽  
Harmonic Mapping ◽  
Boundary Function ◽  
Harmonic Mappings ◽  
C1 Domain ◽  
Sufficient And Necessary Conditions ◽  
Upper Half Plane

Suppose that h is a harmonic mapping of the unit disc onto a C1, ? domain D. Then h is q.c. if and only if it is bi-Lipschitz. In particular, we consider sufficient and necessary conditions in terms of boundary function that h is q.c. We give a review of recent related results including the case when domain is the upper half plane. We also consider harmonic mapping with respect to ? metric on codomain.

scholarly journals On optimal recovery of dirichlet problem from a boundary function known approximately

2015 ◽  
Author(s):  
Е.В. Абрамова
Keyword(s):  
Dirichlet Problem ◽  
Boundary Function ◽  
Optimal Recovery

В работе рассматривается задача о наилучшем (оптимальном) восстановлении решения задачи Дирихле для верхней полуплоскости по точно или приближенно известному преобразованию Фурье граничной функции. Построена серия оптимальных методов восстановления и вычислена соответствующая погрешность восстановления.

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