Spherical Functions and Spectrum of the Laplacian on Semi-homogeneous Trees

In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely-additive measures (distributions) of certain functions depending on the Poisson kernel. We define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to Almansi holds for polymartingales. We then show that on homogeneous trees there are 1-1 correspondences among the space of n th-order polyharmonic functions, the space of n th-order polymartingales, and the space of n -tuples of distributions. Finally, we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions.

This chapter systematically presents the Legendre and associated Legendre functions and spherical harmonics, which should be considered as essential knowledge of geodesy. The approach is to derive a unique and finite solution of Laplace’s equation in the spherical coordinate system based on Fourier’s method of variable separation. The expression of the Earth’s external gravitational potential, along with the gravitational attraction and its gradient, is formulated in terms of a spherical harmonic series. In particular, the azimuthal average of spherical harmonics and the convergence of the spherical harmonic series of a function defined on a sphere, i.e., the Laplace series, are formulated based on fundamental calculus. As an illustrative example of application, the Poisson integral is derived by solving Dirichlet’s boundary value problem outside a sphere using spherical harmonic series.

The K-invariant probability m on F = G/P represents, by means of the square root of the Poisson kernel, a unique solution of the equation Lu + λ 0u = 0 with u(o) = 1. It is the spherical function Φ defined by \(\Phi \left( {g \cdot o} \right) = \int {_K{e^{ - \rho \left( {H\left( {{g^{ - 1}}k} \right)} \right)}}} dk = \int {_\mathcal{F}{P^{1/2}}\left( {x,b} \right)dm\left( b \right),} \) where x = g · o ∈ X, b = kP, and \(P\left( {x,b} \right) = {e^{ - 2\rho \left( {H\left( {{g^{ - 1}}k} \right)} \right)}}\) is the Poisson kernel on X (see §7.21 and § 8.27). It plays a basic role in harmonic analysis on semisimple groups, for example it dominates all the spherical functions associated with the unitary principal series, and is called the Harish-Chandra spherical function (see [G1]). It arose earlier in Chapter VII when determining the limit functions for the Martin compactification at the bottom of the positive spectrum.

LP(#) into Lq(/t), where/z is the Gauss measure, for co real, co2~ p 1 1 < p < q < oo. q l ' This has been extended to imaginary 0) by Beckner [3] (this enabled him to give a sharp version of the Hausdorff-Young inequality) and to complex co by Weissler [16]. Several other proofs using widely different techniques have appeared [9], [6], [14], [111, [8], [1]. Weissler [17] proved that the Poisson integral f-Po~f on the unit circle also is hypercontractive (for the Haar measure) with the above bound on 0). It will be proved in this paper (Section 6) that the same is true for the Poisson integral on a sphere in R s. It is an open problem whether this holds also in higher dimensions. The eigenfunctions of the Mehler transform and the Poisson integral are the Hermite polynomials and the spherical harmonics respectively, which are the ortogonal polynomials for the respective measures. In the present paper we use this as the definition of a family of operators for any probability measure on R d (Section 1). The general and still unsolved problem is to decide when the operators are contractions and, in particular, for which measures this holds with Nelson's conditions on co. It shown (Section 3) that this condition always is necessary, but no general sufficient condition is known. However, several theorems (Section 4) prove the hypercontractivity property for one measure, assuming it for others, and these theorems enable us to prove the result stated above for a sphere. The final section treats multipliers on orthogonal polynomials of complex vari-

Alternative expressions for calculating the prolate spheroidal radial functions of the second kind R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) and their first derivatives with respect to ξ \xi are shown to provide accurate values over wide parameter ranges where the traditional expressions fail to do so. The first alternative expression is obtained from the expansion of the product of R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}(c, \xi ) and the prolate spheroidal angular function of the first kind S m l ( 1 ) ( c , η ) S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right ) in a series of products of the corresponding spherical functions. A similar expression for the radial functions of the first kind was shown previously to provide accurate values for the prolate spheroidal radial functions of the first kind and their first derivatives over all parameter ranges. The second alternative expression for R m l ( 2 ) ( c , ξ ) R_{ml}^{\left ( 2 \right )}\left ( c, \xi \right ) involves an integral of the product of S m l ( 1 ) ( c , η ) S_{ml}^{\left ( 1 \right )}\left ( c, \eta \right ) and a spherical Neumann function kernel. It provides accurate values when ξ \xi is near unity and l − m l - m is not too large, even when c c becomes large and traditional expressions fail. The improvement in accuracy using the alternative expressions is quantified and discussed.

Geomathematics provides a comprehensive summary of the mathematical principles behind key topics in geophysics and geodesy, covering the foundations of gravimetry, geomagnetics and seismology. Theorems and their proofs explain why physical realities in geoscience are the logical mathematical consequences of basic laws. The book also derives and analyzes the theory and numerical aspects of established systems of basis functions; and presents an algorithm for combining different types of trial functions. Topics cover inverse problems and their regularization, the Laplace/Poisson equation, boundary-value problems, foundations of potential theory, the Poisson integral formula, spherical harmonics, Legendre polynomials and functions, radial basis functions, the Biot-Savart law, decomposition theorems (orthogonal, Helmholtz, and Mie), basics of continuum mechanics, conservation laws, modelling of seismic waves, the Cauchy-Navier equation, seismic rays, and travel-time tomography. Each chapter ends with review questions, with solutions for instructors available online, providing a valuable reference for graduate students and researchers.

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

A numerical integration method for analyzing the geomagnetic field is developed from Poisson's integral. A new surface grid, suitable for use with integral analysis, is described. This grid is based on subdivisions of a spherical icosahedron, and its points are almost uniformly spaced over a sphere. This integration method is applied to calculations of field values, field lines, and conjugate points. The results are compared with those of earlier spherical-harmonic analyses by Vestine and Sibley. A comparison is also made between those conjugate points calculated by spherical harmonics from different sets of coefficients derived from various sets of isomagnetic charts.

The subject of analysis on Lie groups comprises an eclectic group of topics which can be treated from many different perspectives. This self-contained text concentrates on the perspective of analysis, to the topics and methods of non-commutative harmonic analysis, assuming only elementary knowledge of linear algebra and basic differential calculus. The author avoids unessential technical discussions and instead describes in detail many interesting examples, including formulae which have not previously appeared in book form. Topics covered include the Haar measure and invariant integration, spherical harmonics, Fourier analysis and the heat equation, Poisson kernel, the Laplace equation and harmonic functions. Perfect for advanced undergraduates and graduates in geometric analysis, harmonic analysis and representation theory, the tools developed will also be useful for specialists in stochastic calculation and the statisticians. With numerous exercises and worked examples, the text is ideal for a graduate course on analysis on Lie groups.

Starting with the computation of the appropriate Poisson kernels, we review, complement, and compare results on drifted Laplace operators in two different contexts: homogeneous trees and the hyperbolic half-plane.

In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, Poisson kernels and spherical functions, i.e., the kernel of the spherical Fourier transform. We introduce and exploit a simple new method of construction of these W‐invariant kernels by alternating sum formulas. We then use the alternating sum representation of these kernels to obtain their asymptotic behavior. We apply our results to the Dyson Brownian Motion on .

A surface reflectance function represents the process of turning irradiance signals into outgoing radiance. Irradiance signals can be represented using low-order basis functions due to their low-frequency nature. Spherical harmonics (SH) have been used to provide such basis. However the incident light at any surface point is defined on the upper hemisphere, full spherical representation is not needed. We propose the use of hemispherical harmonics (HSH) to model images of convex Lambertian objects under distant illumination. We formulate and prove the addition theorem for HSH in order to provide an analytical expression of the reflectance function in the HSH domain. We prove that the Lambertian kernel has a more compact harmonic expansion in the HSH domain when compared to its SH counterpart. We present the approximation of the illumination cone in the HSH domain where the non-negative lighting constraint is forced. Our experiments illustrate that the 1st order HSH outperforms 1st and 2nd order SH in the process of image reconstruction as the number of light sources grows. In addition, we provide empirical justification of the nonnecessity of enforcing the non-negativity constraint in the HSH domain, allowing us to use unconstrained least squares which is faster than the constrained one, while maintaining the same quality of the reconstructed images.

A surface reflectance function represents the process of turning irradiance signals into outgoing radiance. Irradiance signals can be represented using low-order basis functions due to their low-frequency nature. Spherical harmonics (SH) have been used to provide such basis. However the incident light at any surface point is defined on the upper hemisphere; full spherical representation is not needed. We propose the use of hemispherical harmonics (HSH) to model images of convex Lambertian objects under distant illumination. We formulate and prove the addition theorem for HSH in order to provide an analytical expression of the reflectance function in the HSH domain. We prove that the Lambertian kernel has a more compact harmonic expansion in the HSH domain when compared to its SH counterpart. Our experiments illustrate that the 1st order HSH outperforms 1st and 2nd order SH in the process of image reconstruction as the number of light sources grows.

Universal properties of the isotropic Laplace operator on homogeneous trees

ABSTRACTThis article presents the definition of Markov chain indexed by homogeneous trees in Markovian environment. Then we mainly study the strong limit theorems for a Markov chain indexed by homogeneous trees in Markovian environment. We also establish the strong law of large numbers and the Shannon–McMillan theorems for finite Markov chains indexed by a homogeneous tree in a Markovian environment with finite state space. We only prove the results on a Bethe tree and then just state the analogous results on a rooted Cayley tree. There are abundant achievements in research of Markov chains in determined environments, but few results about this topic of Markov chains indexed by trees in random environment. The results of this manuscript are very meaningful to give a good start to establish the strong limit properties for Markov chains indexed by trees in different random environments.