scholarly journals Integral representations of a class of harmonic functions in the half space

2016 ◽  
Vol 260 (2) ◽  
pp. 923-936 ◽  
Author(s):  
Yan Hui Zhang ◽  
Guan Tie Deng ◽  
Tao Qian
Keyword(s):  
Half Space ◽  
Harmonic Functions ◽  
Integral Representations

scholarly journals Solving Integral Representations Problems for the Stationary Schrödinger Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yudong Ren
Keyword(s):  
Schrödinger Equation ◽  
Half Space ◽  
Harmonic Functions ◽  
Schrodinger Equation ◽  
Integral Representations ◽  
Representation Theorems ◽  
Stationary Schrödinger Equation

When solutions of the stationary Schrödinger equation in a half-space belong to the weighted Lebesgue classes, we give integral representations of them, which imply known representation theorems of classical harmonic functions in a half-space.

A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces

1970 ◽  
Vol 22 (4) ◽  
pp. 855-862
Author(s):  
Y. K. Kwon ◽  
L. Sario
Keyword(s):  
Maximum Principle ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Decomposition Theorem ◽  
Higher Dimensions ◽  
Integral Representations ◽  
Classification Theory ◽  
Topological Properties ◽  
Harmonic Decomposition ◽  
Bounded Harmonic Functions

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).

scholarly journals Integral representations for Padé-type operators

2002 ◽  
Vol 2 (2) ◽  
pp. 51-69
Author(s):  
Nicholas J. Daras
Keyword(s):  
Fourier Series ◽  
Explicit Form ◽  
Harmonic Functions ◽  
Integral Operators ◽  
Integral Representations ◽  
Open Disk ◽  
Integral Formulas ◽  
Convergence Results ◽  
Do So

The main purpose of this paper is to consider an explicit form of the Padé-type operators. To do so, we consider the representation of Padé-type approximants to the Fourier series of the harmonic functions in the open disk and of theL p-functions on the circle by means of integral formulas, and, then we define the corresponding Padé-type operators. We are also oncerned with the properties of these integral operators and, in this connection, we prove some convergence results.

Poisson’s integral formula via heat kernels

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 59-64
Author(s):  
Yoichi Miyazaki
Keyword(s):  
Fourier Transform ◽  
Heat Kernel ◽  
Half Space ◽  
Unbounded Domain ◽  
Harmonic Functions ◽  
Integral Formula ◽  
Heat Kernels ◽  
Transform Method ◽  
Fourier Transform Method ◽  
The Fourier Transform

Abstract We give another proof of Poisson’s integral formula for harmonic functions in a ball or a half space by using heat kernels with Green’s formula. We wish to emphasize that this method works well even for a half space, which is an unbounded domain; the functions involved are integrable, since the heat kernel decays rapidly. This method needs no trick such as the subordination identity, which is indispensable when applying the Fourier transform method for a half space.

scholarly journals Inferior Mean of Measures on Curves and Subspaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yevgenya Movshovich
Keyword(s):  
Half Space ◽  
Harmonic Functions ◽  
Upper Bounds ◽  
Finite Boundary ◽  
Positive Harmonic Functions

We obtain new sharp upper bounds of the inferior mean for positive harmonic functions defined by finite boundary measures that lie on curves or subspaces of the boundary of the half-space.

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