Poisson Kernel Research Articles

One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

We consider the generalized Poisson kernel Π q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (−1, 1) and α ∈ ℝ, which is a linear combination of the Poisson kernel $$P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt$$ and the conjugate Poisson kernel $$Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt$$ . The values of the best integral approximation to the kernel Π q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.

Interpolation Wavelets in Boundary Value Problems

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let $L$ be a Schr\"odinger operator of the form $L=-\Delta+V$ acting on $L^2(\mathbb R^n)$ where the nonnegative potential $V$ belongs to the reverse H\"older class $B_q$ for some $q\geq n.$ In this article we will show that a function $f\in L^{2, \lambda}({\mathbb{R}^n}), 0<\lambda<n$ is the trace of the solution of ${\mathbb L}u=-u_{tt}+L u=0, u(x,0)= f(x),$ where $u$ satisfies a Carleson type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-\lambda}\int_0^{r_B}\int_{B(x_B, r_B)} t|\nabla u(x,t)|^2 {dx dt} \leq C <\infty. \end{eqnarray*} Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces $\mathscr{L}_L^{2,\lambda}({\mathbb{R}^n})$ associated to the operator $L$, i.e. $$\mathscr{L}_L^{2,\lambda}(\mathbb{R}^n)= {L}^{2,\lambda}(\mathbb{R}^n). $$ Conversely, this Carleson type condition characterizes all the ${\mathbb L}$-harmonic functions whose traces belong to the space $L^{2, \lambda}({\mathbb{R}^n})$ for all $ 0<\lambda<n$. This extends the previous results of [FJN, DYZ, JXY].

Spherical polyharmonics and Poisson kernels for polyharmonic functions

ABSTRACTWe introduce and develop the notion of spherical polyharmonics, which are a natural generalisation of spherical harmonics. In particular we study the theory of zonal polyharmonics, which allows us, analogously to zonal harmonics, to construct Poisson kernels for polyharmonic functions on the union of rotated balls. We find the representation of Poisson kernels and zonal polyharmonics in terms of the Gegenbauer polynomials. We show the connection between the classical Poisson kernel for harmonic functions on the ball, Poisson kernels for polyharmonic functions on the union of rotated balls, and the Cauchy-Hua kernel for holomorphic functions on the Lie ball.

Asymptotic behavior of Poisson integrals in a cylinder and its application to the representation of harmonic functions

Our first aim in this paper is to deal with asymptotic behavior of Poisson integrals in a cylinder. Next Carleman's formula for harmonic functions in it is also proved. As an application of them, we finally give the integral representation of harmonic functions in a cylinder.

Littlewood-Paley g-function and Radon Transform on the Heisenberg Group

In this paper, we consider Radon transform on the Heisenberg group $\textbf{H}^{n}$, and obtain new inversion formulas via dual Radon transforms and Poisson integrals. We prove that the Radon transform is a unitary operator from Sobelov space $W$ into $L^{2}(\textbf{H}^{n})$. Moreover, we use the Radon transform to define the Littlewood-Paley $g$-function on a hyperplane and obtain the Littlewood-Paley theory.

On a Conformally Invariant Integral Equation Involving Poisson Kernel

We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan–Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class.

A disturbance observer-based robust controller design for systems with right half plane zeros and poles

This paper analytically derives the bandwidth limitations of Disturbance Observer (DOB) when plants have Right Half Plane (RHP) zero(s) and pole(s). If the plant is non-minimum phase, then the bandwidth of DOB should be set at a lower value than its upper bound to improve the robust stability and performance. If the plant is unstable, then the bandwidth of DOB should be set at a higher value than its lower bound to achieve the robust stability. The upper and lower bounds are analytically derived by using Poisson integral formula. It is shown that the bandwidth limitation of DOB is directly related to the locations of the RHP zero(s) and pole(s) and becomes more severe as they get close each other. A minimum phase approximation of the non-minimum phase nominal plant model is proposed by using Genetic Algorithm (GA) to tackle the internal stability problem of the DOB-based robust control systems. Simulation results are given to verify the proposed robust controllers.

The Solution of the Wave Equation in the Class of Complex Numbers by Introducing Hyperbolic Coordinates

The finding of the solution of the wave equation, formulated as the Cauchy problem, does not exhaust all possibilities of the theory. The attempt to examine that one by admitting that the time is an imaginary value is made. So the new curvilinear coordinates, named hyperbolic, are introduced in consideration. They allow for hyperbolic equations to extend a field of searching of solutions to the complex plan and give the possibility to apply powerful Fourier’s method. Due to that, the wave equation takes a form of Laplace’s one in polar coordinates. However, the boundary condition differs from well known Dirichlet problem that in this case looses the sence. The new condition is admitted and it is physically formulated as the description of wave from various inertial systems of coordinates. So the result is obtaining proceeding either of the momentum picture of a wave, made from the moving system of coordinates, or on the oscillogram, developed in time The analytic solution that differs from Poisson integral is deduced and gives the formulas of relativistic addition of velocities for points of wave, observing from different inertial systems. That integral was also formally yielded by using the conform translation. Additionally, in the frequencies field those formulas describe the relativistic Doppler’s effect and the red shift in the wave spectrum. For oscillatory boundary condition the solution of the obtained integral gives a description of the shock waves. The fact, that some formulas of Relativity may be deduced by new way, gives the possibility to explain the relativistic theory proceeding from supposition of waving nature of quantum objects.

Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

The effect of anomalous events on the replicator dynamics with aggregate shocks is considered. The anomalous events are described by a Poisson integral, where this stochastic forcing term is added to the fitness of each agent. Contrary to previous models, this noise is assumed to be correlated across the population. A formula to calculate a closed form solution of the long run behavior of a two strategy game will be derived. To assist with the analysis of a two strategy game, the stochastic Lyapunov method will be applied. For a population with a general number of strategies, the time averages of the dynamics will be shown to converge to the Nash equilibria of a relevant modified game. In the context of the modified game, the almost sure extinction of a dominated pure strategy will be derived. As the dynamic is quite complex, with respect to the original game a pure strict Nash equilibrium and an interior evolutionary stable strategy will be considered. Respectively, conditions for stochastically stability and the positive recurrent property will be derived. This work extends previous results on the replicator dynamics with aggregate shocks.

Approximation of classes of Poisson integrals by Fejer sums

For upper bounds of the deviations of Fejer sums taken over classes of periodic functions that admit analytic extensions to a fixed strip of the complex plane, we obtain asymptotic equalities. In certain cases, these equalities give a solution of the corresponding Kolmogorov-Nikolsky problem.

Generalized Poisson integral and sharp estimates for harmonic and biharmonic functions in the half-space

A representation for the sharp coefficient in a pointwise estimate for the gradient of a generalized Poisson integral of a function f on ℝn−1 is obtained under the assumption that f belongs to Lp. It is assumed that the kernel of the integral depends on the parameters α and β. The explicit formulas for the sharp coefficients are found for the cases p = 1, p = 2 and for some values of α, β in the case p = ∞. Conditions ensuring the validity of some analogues of the Khavinson’s conjecture for the generalized Poisson integral are obtained. The sharp estimates are applied to harmonic and biharmonic functions in the half-space.

Asymptotic Expansions of Upper Bounds of Deviations of Functions of Class Wr from Their Generalized Poisson Integrals

Asymptotic Expansions of Upper Bounds of Deviations of Functions of Class Wr from Their Generalized Poisson Integrals

Connection of the B¨ottcher equation with the parametrized Poisson integral

Connection of the B¨ottcher equation with the parametrized Poisson integral

Lebesgue-type inequalities for the Fourier sums on classes of generalized Poisson integrals

Lebesgue-type inequalities for the Fourier sums on classes of generalized Poisson integrals

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation $\Delta_{h}u(x)=\psi(x)$, where $\psi\in L^{\infty}(\mathbb{B}^{n}, \mathbb{R}^n)$ and \[ \Delta_{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \] is the hyperbolic Laplace operator in the $n$-dimensional space $\mathbb{R}^n$ for $n\ge 2$. We show that if $n\geq 3$ and $u\in C^{2}(\mathbb{B}^{n},\mathbb{R}^n) \cap C(\overline{\mathbb{B}^{n}},\mathbb{R}^n )$ is a solution to the hyperbolic Poisson equation, then it has the representation $u=P_{h}[\phi]-G_{ h}[\psi]$ provided that $u\mid_{\mathbb{S}^{n-1}}=\phi$ and $\int_{\mathbb{B}^{n}}(1-|x|^{2})^{n-1} |\psi(x)|\,d\tau(x)<\infty$. Here $P_{h}$ and $G_{h}$ denote Poisson and Green integrals with respect to $\Delta_{h}$, respectively. Furthermore, we prove that functions of the form $u=P_{h}[\phi]-G_{h}[\psi]$ are Lipschitz continuous.

Explicit formulas for the Dunkl dihedral kernel and the (κ,a)-generalized Fourier kernel

In this paper, a new method is developed to obtain explicit and integral expressions for the kernel of the (κ,a)-generalized Fourier transform for κ=0. In the case of dihedral groups, this method is also applied to the Dunkl kernel as well as the Dunkl Bessel function. The method uses the introduction of an auxiliary variable in the series expansion of the kernel, which is subsequently Laplace transformed. The kernel in the Laplace domain takes on a much simpler form, by making use of the Poisson kernel. The inverse Laplace transform can then be computed using the generalized Mittag–Leffler function to obtain integral expressions. In case the parameters involved are integers, explicit formulas are obtained using partial fraction decomposition. New bounds for the kernel of the (κ,a)-generalized Fourier transform are obtained as well.

A Class of $\alpha$-Carleson Measures

In the present paper, we introduce a class of $\alpha$-Carleson measures $\mathcal{C}_{\alpha,v}(\mathbb{R}^{n+1}_+)$, which is called by the vanishing $\alpha$-Carleson measures. We prove that $\mathcal{C}_{1/p,v}(\mathbb{R}^{n+1}_+)$ is just a predual of the tent space $\widetilde{T}_{\infty}^p$ ($0 \lt p \lt 1$). Furthermore, we construct the $\alpha$-Carleson measures and the vanishing $\alpha$-Carleson measures by the Campanato functions and its a subclass, respectively. Moreover, a characterization of the vanishing $\alpha$-Carleson measure by the compactness of Poisson integral is given in this paper. Finally, as some applications, we give the $(L^{2/\alpha},L^2)$ boundedness and compactness for some paraproduct operators.

Rational Function Operators from Poisson Integrals

AbstractIn this paper, we construct two classes of rational function operators using the Poisson integrals of the function on the whole real axis. The convergence rates of the uniform and mean approximation of such rational function operators on the whole real axis are studied.

Approximative Properties of Biharmonic Poisson Integrals on Hölder Classes

We obtain asymptotic expansions for the approximations of functions from the Holder class by biharmonic Poisson integrals in the uniform and integral metrics.