On zeros, singular boundary functions, and modules of angular boundary values for one class of functions analytic in a half-plane

2004 ◽  
Vol 56 (6) ◽  
pp. 1015-1022
Author(s):  
B. V. Vinnitskii ◽  
V. L. Sharan
Keyword(s):  
Half Plane ◽  
Singular Boundary ◽  
Boundary Values ◽  
Boundary Functions ◽  
Angular Boundary ◽  
Class Of Functions

A Short Proof of an Interpolation Theorem

1974 ◽  
Vol 17 (1) ◽  
pp. 127-128 ◽  
Author(s):  
Edward Hughes
Keyword(s):  
Half Plane ◽  
Complex Plane ◽  
Simple Proof ◽  
Real Line ◽  
Short Proof ◽  
Interpolation Theorem ◽  
Operator Interpolation ◽  
Upper Half Plane ◽  
Positive Functions ◽  
Class Of Functions

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

scholarly journals Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

2018 ◽  
pp. 1-12
Author(s):  
O. D. Algazin
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Mixed Type ◽  
Polynomial Growth ◽  
Polynomial Solution ◽  
Transform Method ◽  
Polynomial Solutions ◽  
Tricomi Equation ◽  
Equation Of Mixed Type ◽  
Class Of Functions

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

scholarly journals A Note on the Displacement Problem of Elastostatics with Singular Boundary Values

Axioms ◽  
2019 ◽  
Vol 8 (2) ◽  
pp. 46 ◽  
Author(s):  
Alfonsina Tartaglione
Keyword(s):  
Elastic Body ◽  
Function Class ◽  
Singular Boundary ◽  
Regular Boundary ◽  
Boundary Values ◽  
Exterior Domains ◽  
Displacement Problem ◽  
Natural Function ◽  
Linear Elastostatics ◽  
Class C

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum a is considered. Precisely, if the elastic body is represented by a domain of class C k ( k ≥ 2 ) of R 3 and a ∈ W 2 − k − 1 / q , q ( ∂ Ω ) , q ∈ ( 1 , + ∞ ) , then it is proved that there exists a solution which is of class C ∞ in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class.

scholarly journals Nonsteady frictional heating on a sliding contact

2018 ◽  
Vol 226 ◽  
pp. 03030
Author(s):  
Vladimir B. Zelentsov ◽  
Boris I. Mitrin
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Contact Problem ◽  
Half Plane ◽  
Frictional Heating ◽  
Contact Stresses ◽  
Sliding Contact ◽  
Singular Equation ◽  
Static Contact ◽  
Class Of Functions

We consider quasi-static contact problem on frictional heating on a sliding contact of a rotating rigid cylinder and a half-plane. The cylinder is pressed towards the half-plane material. The problem is reduced to solution of a singular integral equation with respect to contact stresses. Solution of the singular equation is looked for in a class of functions limited on the edge, with two additional conditions to determine timedependent boundaries of the contact area. Temperature at the contact and inside the half-plane is determined in terms of contact stresses.

SOME PROPERTIES OF BLASCHKE TYPE PRODUCTS FOR THE HALF-PLANE

2020 ◽  
Vol 54 (2 (252)) ◽  
pp. 101-107
Author(s):  
G.V. Mikayelyan ◽  
V.S. Petrosyan
Keyword(s):  
Half Plane ◽  
Boundary Values ◽  
Logarithmic Means

In this paper we obtain balance formulas for the logarithmic means of Blaschke type functions and investigate their boundary values.

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