Poisson Kernel Research Articles

Radial boundary values of Poisson integrals on infinite-dimensional balls

We consider a Gelfand triple $E'\rightarrow H\rightarrow E$ , so that E is a separable complex Banach space with dual $E'$ , and H is its dense Hilbert subspace. We investigate the problem of analytic extensions on an open ball $\mathcal{Q}\subset E'$ and their radial boundary values in the Hardy spaces $\mathcal{H}_{\mu}^{p}$ ( $1\le p\le\infty$ ) using the Poisson integrals on the unitary group $U(\infty)$ over H endowed with an invariant probability measure μ. For this purpose, we construct a Poisson-type kernel with the help of the symmetric Fock space Γ generated by H and prove that the set of radial boundary values of these analytic functions entirely coincides with $\mathcal{H}_{\mu}^{p}$ .

An application of the inequality for modified Poisson kernel

As an application of an inequality for modified Poisson kernel obtained by Qiao and Deng (Bull. Malays. Math. Sci. Soc. (2) 36(2):511-523, 2013), we give the generalized solution of the Dirichlet problem with arbitrary growth data.

Experimental investigation on the uniqueness of a center of a body

The object of our investigation is a point that gives the maximum value of a potential with a strictly decreasing radially symmetric kernel. It defines a center of a body in Rm. When we choose the Riesz kernel or the Poisson kernel as the kernel, such centers are called a radial center or an illuminating center, respectively. The existence of a center is easily shown but the uniqueness does not always hold. Sufficient conditions of the uniqueness of a center have been studied by some researchers. The main results in this paper are some new sufficient conditions for the uniqueness of a center of a body.

Proofs of the Density Theorem and Fatou’s Radial Limit Theorem Using the Poisson Integral

Using only the Poisson integral and elementary convergence theorems, we prove the well-known Density theorem and Fatou’s radial limit theorem.

The Dirichlet problem for standard weighted Laplacians in the upper half plane

In this paper the Dirichlet problem for a class of standard weighted Laplace operators in the upper half plane is solved by means of a counterpart of the classical Poisson integral formula. Boundary limits and representations of the associated solutions are studied within a framework of weighted spaces of distributions. Special attention is given to the development of a suitable uniqueness theory for the Dirichlet problem under appropriate growth constraints at infinity.

Variational discretization for the planar Lotka–Volterra equations in the Birkhoffian sense

In this paper, we derive the variational characterization of the planar Lotka–Volterra equations in the Birkhoffian sense and ulteriorly construct variational integrators for the group of equations. The planar Lotka–Volterra equations turn out to admit a Birkhoffian representation and consequently can be discretized according to the discrete Birkhoffian equations. By means of the transformation theory of the Birkhoffian equations if necessary, efficient variational integrators of the Lotka–Volterra equations can be obtained. These variational integrators, compared with traditional difference schemes as well as Poisson integrators, have better numerical performance in terms of stability, accuracy and preservation of conserved quantities, demonstrated by numerical results.

On the wave representation of hyperbolic, elliptic, and parabolic Eisenstein series

We develop a unified approach to the construction of the hyperbolic and elliptic Eisenstein series on a finite volume hyperbolic Riemann surface. Specifically, we derive expressions for the hyperbolic and elliptic Eisenstein series as integral transforms of the kernel of a wave operator. Established results in the literature relate the wave kernel to the heat kernel, which admits explicit construction from various points of view. Therefore, we obtain a sequence of integral transforms which begins with the heat kernel, obtains a Poisson and wave kernel, and then yields the hyperbolic and elliptic Eisenstein series. In the case of a non-compact finite volume hyperbolic Riemann surface, we finally show how to express the parabolic Eisenstein series in terms of the integral transform of a wave kernel.

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

In this article, motivated by the classic Hadamard factorization theorem about an entire function of finite order in the complex plane, we firstly prove that a harmonic function whose positive part satisfies some growth conditions, can be represented by its integral on the boundary of the half space. By using Nevanlinna's representation of harmonic functions and the modified Poisson kernel of the half space, we further prove a representation formula through integration against a certain measure on the boundary hyperplane for harmonic functions not necessarily continuous on the boundary hyperplane whose positive parts satisfy weaker growing conditions than the first question. The result is further generalized by involving a parameter m dealing with the singularity at the infinity.

SELF-GRAVITATIONAL FORCE CALCULATION OF SECOND-ORDER ACCURACY FOR INFINITESIMALLY THIN GASEOUS DISKS IN POLAR COORDINATES

Investigating the evolution of disk galaxies and the dynamics of proto-stellar disks can involve the use of both a hydrodynamical and a Poisson solver. These systems are usually approximated as infinitesimally thin disks using two- dimensional Cartesian or polar coordinates. In Cartesian coordinates, the calcu- lations of the hydrodynamics and self-gravitational forces are relatively straight- forward for attaining second order accuracy. However, in polar coordinates, a second order calculation of self-gravitational forces is required for matching the second order accuracy of hydrodynamical schemes. We present a direct algorithm for calculating self-gravitational forces with second order accuracy without artifi- cial boundary conditions. The Poisson integral in polar coordinates is expressed in a convolution form and the corresponding numerical complexity is nearly lin- ear using a fast Fourier transform. Examples with analytic solutions are used to verify that the truncated error of this algorithm is of second order. The kernel integral around the singularity is applied to modify the particle method. The use of a softening length is avoided and the accuracy of the particle method is significantly improved.

Chordal Komatu-Loewner equation and Brownian motion with darning in multiply connected domains

Let D = H ∖ ⋃ k = 1 N C k D={\mathbb H} \setminus \bigcup _{k=1}^N C_k be a standard slit domain where H \mathbb H is the upper half-plane and C k C_k , 1 ≤ k ≤ N 1\leq k\leq N , are mutually disjoint horizontal line segments in H \mathbb {H} . Given a Jordan arc γ ⊂ D \gamma \subset D starting at ∂ H , \partial {\mathbb H}, let g t g_t be the unique conformal map from D ∖ γ [ 0 , t ] D\setminus \gamma [0,t] onto a standard slit domain D t D_t satisfying the hydrodynamic normalization. We prove that g t g_t satisfies an ODE with the kernel on its right-hand side being the complex Poisson kernel of the Brownian motion with darning (BMD) for D t D_t , generalizing the chordal Loewner equation for the simply connected domain D = H . D={\mathbb H}. Such a generalization has been obtained by Y. Komatu in the case of circularly slit annuli and by R. O. Bauer and R. M. Friedrich in the present chordal case, but only in the sense of the left derivative in t t . We establish the differentiability of g t g_t in t t to make the equation a genuine ODE. To this end, we first derive the continuity of g t ( z ) g_t(z) in t t with a certain uniformity in z z from a probabilistic expression of ℑ g t ( z ) \Im g_t(z) in terms of the BMD for D D , which is then combined with a Lipschitz continuity of the complex Poisson kernel under the perturbation of standard slit domains to get the desired differentiability.

Poisson and fourier transforms for tensor products

For the group G = SL(2,R), we write out explicitly differential operators intertwining irreducible finite-dimensional representations Tk of G with tensor products Tl ⊗ Tm (we call them Poisson and Fourier transforms); we also describe an analogue of harmonic analysis and write explicit expressions for compositions of these transforms with Lie operators of the overgroup G×G. The constructions are based on a differential-difference relation for the Poisson kernel.

Positive Solutions of a Semilinear Elliptic Equation with Singular Dirichlet Boundary Data

The purpose of this paper is to construct positive solutions of the semilinear elliptic equation −Δu = up in ℝ+N with a singular Dirichlet boundary condition. We show that for p > (N+1)=(N−1) there exists a positive singular solution which behaves like |x|−2/(p−1) as |x| → 0 and like the Poisson kernel as |x| → ∞.

A solution to Sheil-Small’s harmonic mapping problem for polygons

The problem of mapping the interior of a Jordan polygon univalently by the Poisson integral of a step function was posed by T. Sheil-Small (1989). We describe a simple solution using “ear clipping” from computational geometry.

Movement of centers with respect to various potentials

We investigate a potential with a radially symmetric and strictly decreasing kernel depending on a parameter. We regard the potential as a function defined on the upper half-space R m × ( 0 , + ∞ ) \mathbb {R}^m \times (0,+\infty ) and study some geometric properties of its spatial maximizer. To be precise, we give some sufficient conditions for the uniqueness of a maximizer of the potential and study the asymptotic behavior of the set of maximizers. Using these results, we imply geometric properties of some specific potentials. In particular, we consider applications for the solution of the Cauchy problem for the heat equation, the Poisson integral (including a solid angle) and r α − m r^{\alpha -m} -potentials.

On locality of Radon to Riesz transform

In this paper we present a novel approach to locally compute the Riesz transform from the knowledge of the Radon transform. Previous implementations of the Riesz transform are based on the Fourier or the Radon transforms and their inversion formulae, and therefore needs for the knowledge of the function or its Radon data on the whole domain. More recent attempts on rectangular domains involves convolutions with the Poisson kernel and local derivatives. On the other hand, starting from the links between the Riesz and the Radon transforms, we address in this paper a new local Radon based Riesz formula in the general n-dimensional case, for even n. The advantage of this formula, local in the Radon space, is pointed out in the bidimensional case, where we provide a new local Radon based Riesz algorithm, and conduct numerical tests for the estimation of the Riesz transform on convex sets, from truncated Radon data. Finally we study the robustness to noise of the current approach.

Приближение классов интегралов Пуассона суммами Фейера

Получена ассимптотическая формула для верхних граней уклонений сумм Фейера на классах интегралов Пуассона. В ряде случаев полученное соотношение обеспечивает решение задачи Колмогорова–Никольского для сумм Фейера и классов интегралов Пуассона.

Boundary measures, generalized Gauss–Green formulas, and mean value property in metric measure spaces

We study mean value properties of harmonic functions in metric measure spaces. The metric measure spaces we consider have a doubling measure and support a (1, 1)-Poincaré inequality. The notion of harmonicity is based on the Dirichlet form defined in terms of a Cheeger differentiable structure. By studying fine properties of the Green function on balls, we characterize harmonic functions in terms of a mean value property. As a consequence, we obtain a detailed description of Poisson kernels. We shall also obtain a Gauss–Green type formula for sets of finite perimeter which posses a Minkowski content characterization of the perimeter. For the Gauss–Green formula we introduce a suitable notion of the interior normal trace of a regular ball.

Kinetic models in n-dimensional Euclidean spaces: From the Maxwellian to the Poisson kernel.

In this work, minimal kinetic theories based on unconventional entropy functions, H∼ln f (Burg entropy) for 2D and H∼f(1-2/n) (Tsallis entropy) for nD with n≥3, are studied. These entropy functions were originally derived by Boghosian et al. [Phys. Rev. E 68, 025103 (2003)] as a basis for discrete-velocity and lattice Boltzmann models for incompressible fluid dynamics. The present paper extends the entropic models of Boghosian et al. and shows that the explicit form of the equilibrium distribution function (EDF) of their models, in the continuous-velocity limit, can be identified with the Poisson kernel of the Poisson integral formula. The conservation and Navier-Stokes equations are recovered at low Mach numbers, and it is shown that rest particles can be used to rectify the speed of sound of the extended models. Fourier series expansion of the EDF is used to evaluate the discretization errors of the model. It is shown that the expansion coefficients of the Fourier series coincide with the velocity moments of the model. Employing two-, three-, and four-dimensional (2D, 3D, and 4D) complex systems, the real velocity space is mapped into the hypercomplex spaces and it is shown that the velocity moments can be evaluated, using the Poisson integral formula, in the hypercomplex space. For the practical applications, a 3D projection of the 4D model is presented, and the existence of an H theorem for the discrete model is investigated. The theoretical results have been verified by simulating the following benchmark problems: (1) the Kelvin-Helmholtz instability of thin shear layers in a doubly periodic domain and (2) the 3D flow of incompressible fluid in a lid-driven cubic cavity. The present results are in agreement with the previous works, while they show better stability of the proposed kinetic model, as compared with the BGK type (with single relaxation time) lattice Boltzmann models.

Analytic properties of complex Hermite polynomials

We study the complex Hermite polynomials { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} in some detail, establish operational formulas for them and prove a Kibble-Slepian type formula, which extends the Poisson kernel for these polynomials. Positivity of the associated kernels is discussed. We also give an infinite family of integral operators whose eigenfunctions are { H m , n ( z , z ¯ ) } \{H_{m,n}(z,\bar z)\} . Some inverse relations are also given. We give a two dimensional moment representation for H m , n ( z , z ¯ ) H_{m,n}(z,\bar z) and evaluate several related integrals. We also introduce bivariate Appell polynomials and prove that { H m , n ( z , z ¯ ) } \{H_{m,n}(z, \bar z)\} are the only bivariate orthogonal polynomials of Appell type.

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation Lλu=0 on (0,∞)×(0,∞), where Lλ=∂t2+∂x2−λ(λ−1)x2, with λ>1. In this paper we characterize the solutions of Lλu=0 on (0,∞)×(0,∞) representable by Bessel–Poisson integrals of BMO-functions as the ones satisfying certain Carleson properties.