Poisson Kernel Research Articles

Correction to: Approximation of classes of Poisson integrals by incomplete Fejér means

Correction to: Approximation of classes of Poisson integrals by incomplete Fejér means

Carleson Measures on Locally Finite Trees

We provide a characterization of Carleson measures on locally finite trees. This characterization establishes the connection between Carleson measures and the boundedness of a suitable Poisson integral between Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-spaces. Additionally, when the tree has bounded degree, we investigate the relationship between Carleson measures and BMO functions defined on the boundary of the tree.

On sharp heat kernel estimates in the context of Fourier–Dini expansions

We prove sharp estimates of the heat kernel associated with Fourier–Dini expansions on (0,1) equipped with Lebesgue measure and the Neumann condition imposed on the right endpoint. Then we give several applications of this result including sharp bounds for the corresponding Poisson and potential kernels, sharp mapping properties of the maximal heat semigroup and potential operators and boundary convergence of the Fourier–Dini semigroup.

Correction to: Approximation of classes of Poisson integrals by incomplete Fejér means

Correction to: Approximation of classes of Poisson integrals by incomplete Fejér means

A kind regularization method with Poisson kernel to solve Cauchy problem of Schrödinger equation

In this work, a precise regularization technique based on the mollification method is proposed to solve the ill-posed Schr o ¨ dinger Cauchy problem with potential-free field. In the sense of Hadamard, this problem is severely ill-posed. Using the mollification regularization method based on Poisson kernel, some new error estimates are obtained under the priori and posteriori parameter selection rules. The regular approximate solutions are created, and the convergence evidences are provided. The good performance and high accuracy of this technique are demonstrated through various examples. The solution works well and is resistant to data disruption noise.

Kuramoto variables as eigenvalues of unitary matrices.

We generalize the Kuramoto model by interpreting the N variables on the unit circle as eigenvalues of a N-dimensional unitary matrix U in three versions: general unitary, symmetric unitary, and special orthogonal. The time evolution is generated by N^{2} coupled differential equationsfor the matrix elements of U, and synchronization happens when U evolves into a multiple of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that are so useful in quantum transport, and we prove it in the case of identical natural frequencies. When the coupling constant is a matrix, we find some surprising new dynamical behaviors.

Path Dynamics of Time-Changed Lévy Processes: A Martingale Approach

Lévy processes time-changed by inverse subordinators have been intensively studied in the last years. Their importance in connection with non-local operators and semi-Markov dynamics is well understood, but, in our view, several questions remain open concerning the probabilistic structure of such processes. The time-changed Lévy processes are particularly useful to describe complex systems with fractional and/or anomalous dynamics. The purpose of our work is to analyze the features of the sample paths of such processes, focusing on a martingale-based approach. We introduce the fractional Poisson random measure as the main tool for dealing with the jump component of time-changed càdlàg processes. Further, the fractional random measure is an interesting and novel topic in itself, and thus, it is thoroughly analyzed in the paper. A central role in our analysis is then played by fractional Poisson integrals (involving the aforementioned fractional Poisson measure) which allow a useful description of the random jumps. We investigate these stochastic integrals and the martingale property of their compensated counterpart. Therefore, we are able to obtain a semimartingale representation of time-changed processes analogous to the celebrated Lévy–Itô decomposition. Finally, an approximation scheme of such random processes will be discussed.

On the Dirichlet problem for fractional Laplace equation on a general domain

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems.

Approximation of classes of Poisson integrals by rectangular Fejér means

The article is devoted to the problem of approximation of classes of periodic functions by rectangular linear means of Fourier series. Asymptotic equalities are found for upper bounds of deviations in the uniform metric of rectangular Fejér means on classes of periodic functions of several variables generated by sequences that tend to zero at the rate of geometric progression. In one-dimensional cases, these classes consist of Poisson integrals, namely functions that can be regularly extended in the fixed strip of a complex plane.

On some approximate properties of biharmonic Poisson integrals in the integral metric

This paper is devoted to solving one of the extremal problems in the theory of approximation of functional classes by linear methods of summation of the Fourier series in the integral metric, namely, approximation of classes $L^{\psi}_{\beta, 1}$ by biharmonic Poisson integrals. As a result of the research, we have found the asymptotic equalities for the approximation values of classes of $(\psi, \beta)$-differentiable functions by biharmonic Poisson integrals, that is, have found solutions of the Kolmogorov-Nikol'skii problem for biharmonic Poisson integrals on classes $L^{\psi}_{\beta, 1}$ in the integral metric.

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

On approximation of functions from Hölder classes by biharmonic Poisson integrals defined in the upper half-plane

On approximation of functions from Hölder classes by biharmonic Poisson integrals defined in the upper half-plane

On the Coincidence between Campanato Functions and Lipschitz Functions: A New Approach via Elliptic PDES

Abstract Let $({\mathcal{M}},d,\mu)$ be the metric measure space with a Dirichlet form $\mathscr{E}$. In this paper, we obtain that the Campanato function and the Lipschitz function do always coincide. Our approach is based on the harmonic extension technology, which extends a function u on ${\mathcal{M}}$ to its Poisson integral Ptu on ${\mathcal{M}}\times\mathbb{R}_+$. With this tool in hand, we can utilize the same Carleson measure condition of the Poisson integral to characterize its Campanato/Lipschitz trace, and hence, they are equivalent to each other. This equivalence was previously obtained by Macías–Segovia [Adv. Math., 1979]. However, we provide a new proof, via the boundary value problem for the elliptic equation. This result indicates the famous saying of Stein–Weiss at the beginning of Chapter II in their book [Princeton Mathematical Series, No. 32, 1971].

Solution one-dimensional problems of heat conduction with convection using Poisson integral

There is the relevant problem of developing high-viscosity oil in the oil and gas industry. One of the ways increasing its production is the thermal method. It is necessary to combine hydrodynamic equations with the thermal conductivity equation for simulation of filtration processes. The resulting mathematical models are solved by using different numerical methods. The accuracy and convergence of such algorithms is rarely tested. One way to carry out this check is to simulate problems that can be solved analytically in particular cases. For example, Fourier series are used for simple boundary conditions. Another method is usage the Poisson integral. This method is more convenient for comparing results than the Fourier method, because with the same accuracy requires significantly less calculations. But the Poisson method requires knowledge of the initial condition over the entire space, and in real problems it is usually given in a limited area. In this paper we propose an algorithm for extending the initial conditions to the entire space. In this way, two heat conduction problems are solved taking into account convection using the Poisson integral. It is shown that the accuracy is comparable to the Fourier method, but with fewer calculations.

Polyharmonic potential theory on the Poincaré disk

We consider the open unit disk D equipped with the hyperbolic metric and the associated hyperbolic Laplacian L. For λ∈C and n∈N, a λ-polyharmonic function of order n is a function f:D→C such that (L−λI)nf=0. If n=1, one gets λ-harmonic functions. Based on a Theorem of Helgason on the latter functions, we prove a boundary integral representation theorem for λ-polyharmonic functions. For this purpose, we first determine nth-order λ-Poisson kernels. Subsequently, we introduce the λ-polyspherical functions and determine their asymptotics at the boundary ∂D, i.e., the unit circle. In particular, this proves that, for eigenvalues not in the interior of the L2-spectrum, the zeroes of these functions do not accumulate at the boundary circle. Hence the polyspherical functions can be used to normalise the nth-order Poisson kernels. By this tool, we extend to this setting several classical results of potential theory: namely, we study the boundary behaviour of λ-polyharmonic functions, starting with Dirichlet and Riquier type problems and then proceeding to Fatou type admissible boundary limits.

Numerical methods in Poisson geometry and their application to mechanics

We recall the question of geometric integrators in the context of Poisson geometry and explain their construction. These Poisson integrators are tested in some mechanical examples. Their properties are illustrated numerically and compared to traditional methods.

Asymptotic estimates for deviations of Fejér means on Poisson integrals

The work is devoted to the investigation of problem of approximation of continuous periodic functions of a real variable by trigonometric polynomials generated by linear methods of summation of Fourier series. The simplest example of the process of linear approximation of periodic functions is approximation of functions by partial sums of their Fourier series. However, sequences of partial Fourier sums are not uniformly convergent on the whole class of continuous periodic functions. Therefore, numerous studies are devoted to the study of approximation properties of approximating methods, which are generated by different transformations of sequences of partial sums of the Fourier series and allow obtaining sequences of trigonometric polynomials that are uniformly convergent for the entire class of continuous functions. In particular, Fejér means have been intensively studied in recent decades. One of the important tasks in this direction is study of the asymptotic behavior of the upper bounds of the deviations of trigonometric polynomials for a fixed class of periodic functions. The aim of the work is to systematize the known results concerning the approximation of classes of periodic functions of high smoothness by arithmetic means of Fourier sums, and to present new facts obtained for a more general case. In paper we investigate the asymptotic behavior of upper bounds on classes of Poisson integrals of periodic functions of real variable of deviations of linear means of Fourier series, which are constructed using the Fejér summation method. We study the classes consist of analytic functions of a real variable, which can be regularly extended into the corresponding strip of the complex plane. In the work, asymptotic inequalities for the upper bounds of the deviations of the Fejér means on the class of Poisson integrals were found. In certain case of parameters defining the class, the obtained formulas coincide with previously known asymptotic equalities.

Resonant scattering by a loop: the Wigner delay time and Poisson’s kernel


 
 
 The resonances of a loop pierced by a magnetic field are analized in terms of the scattering matrix phase, the Wigner delay time and its relation to Poisson’s kernel. Except for specific values of the magnetic flux, the resonances appear overlapped by pairs due to the broken degeneracy. Although it is well known that the Poisson kernel describes how the phase is distributed in the Argand plane, we demonstrate that Poisson’s kernel coincides with the reciprocal of the Wigner delay time, thus providing a novel interpretation of this quantity. The distribution of the Wigner delay time is also determined, it exhibits explicitly the effect of the magnetic flux, contrary to what happens to the distribution of the phase.
 
 

Designing Poisson Integrators Through Machine Learning

This paper presents a general method to construct Poisson integrators, i.e., integrators that preserve the underlying Poisson geometry. We assume the Poisson manifold is integrable, meaning there is a known local symplectic groupoid for which the Poisson manifold serves as the set of units. Our constructions build upon the correspondence between Poisson diffeomorphisms and Lagrangian bisections, which allows us to reformulate the design of Poisson integrators as solutions to a certain PDE (Hamilton-Jacobi). The main novelty of this work is to understand the Hamilton-Jacobi PDE as an optimization problem, whose solution can be easily approximated using machine learning related techniques. This research direction aligns with the current trend in the PDE and machine learning communities, as initiated by Physics-Informed Neural Networks, advocating for designs that combine both physical modeling (the Hamilton-Jacobi PDE) and data.

Majorization in Poisson and Bode Type Integrals for Multivariable Systems: Evenness and $k$-Waterbed Effect

Majorization in Poisson and Bode Type Integrals for Multivariable Systems: Evenness and $k$-Waterbed Effect