Theory Of Spherical Functions Research Articles

Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function.

Short Proofs of Explicit Formulas to Boundary Value Problems for Polyharmonic Equations Satisfying Lopatinskii Conditions

This paper deals with Lopatinskii type boundary value problem (bvp) for the (poly) harmonic differential operators. In the case of Robin bvp for the Laplace equation in the ball B1 a Green function is constructed in the cases c>0, c∉−N, where c is the coefficient in front of u in the boundary condition ∂u∂n+cu=f. To do this a definite integral must be computed. The latter is possible in quadratures (elementary functions) in several special cases. The simple proof of the construction of the Green function is based on some solutions of the radial vector field equation Λu+cu=f. Elliptic boundary value problems for Δmu=0 in B1 are considered and solved in Theorem 2. The paper is illustrated by many examples of bvp for Δu=0, Δ2u=0 and Δ3u=0 in B1 as well as some additional results from the theory of spherical functions are proposed.

Radon Transforms in Hyperbolic Spaces and Their Discrete Counterparts

In the hyperbolic disc (or more generally in real hyperbolic spaces) we consider the horospherical Radon transform R and the geodesic Radon transform X. Composition with their respective dual operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree T, separately studied as acting on functions on the vertices or on the edges. This leads to a new theory of spherical functions and Radon inversion on the edges of a tree.

Multiplicative convolution of real asymmetric and real anti-symmetric matrices

AbstractThe singular values of products of standard complex Gaussian random matrices, or sub-blocks of Haar distributed unitary matrices, have the property that their probability distribution has an explicit, structured form referred to as a polynomial ensemble. It is furthermore the case that the corresponding bi-orthogonal system can be determined in terms of Meijer G-functions, and the correlation kernel given as an explicit double contour integral. It has recently been shown that the Hermitised product {X_{M}\cdots X_{2}X_{1}AX_{1}^{T}X_{2}^{T}\cdots X_{M}^{T}}, where each {X_{i}} is a standard real Gaussian matrix and A is real anti-symmetric, exhibits analogous properties. Here we use the theory of spherical functions and transforms to present a theory which, for even dimensions, includes these properties of the latter product as a special case. As an example we show that the theory also allows for a treatment of this class of Hermitised product when the {X_{i}} are chosen as sub-blocks of Haar distributed real orthogonal matrices.

Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type

Let ( M , g ) be a globally symmetric space of noncompact type, of arbitrary rank, and Δ its Laplacian. We introduce a new method to analyze Δ and the resolvent ( Δ - σ ) - 1 ; this has origins in quantum N-body scattering, but is independent of the ‘classical’ theory of spherical functions, and is analytically much more robust. We expect that, suitably modified, it will generalize to locally symmetric spaces of arbitrary rank. As an illustration of this method, we prove the existence of a meromorphic continuation of the resolvent across the continuous spectrum to a Riemann surface multiply covering the plane. We also show how this continuation may be deduced using the theory of spherical functions. In summary, this paper establishes a long-suspected connection between the analysis on symmetric spaces and N-body scattering.

Spherical Harmonic Analysis for Spinors on H n (C)

Recent results on the harmonic analysis of spinor fields on the complex hyperbolic space H n (C) are reviewed. We discuss the action of the invariant differential operators on the Poisson transforms, the theory of spherical functions and the spherical transform. The inversion formula, the Paley-Wiener theorem, and the Plancherel theorem for the spherical transform are obtained by reduction to Jacobi analysis on L 2 (R). Mathematics Subject Classifications (2000): 22E30, 22E46, 33C80, 43A85, 43A90.

Spherical Functions on Ol'shanskii Spaces

Ol'shanskii spaces for semisimple groups are special cases of ordered symmetric spaces. The theory of spherical functions on semisimple Ol'shanskii spaces is extended to general Ol'shanskii spaces. The detailed structure theory for Lie algebras with invariant cones is used to generalize geometric results for Ol'shanskii spaces. Spherical functions are then defined via a set of integral equations using a Volterra algebra that consists ofG-invariant kernels satisfying a causality condition. It is shown that the resulting integral spherical formulas suffice to build a spherical Laplace transform and an inversion formula for this transform.

Spherical functions on orthogonal groups

Let G be a p-adic connected reductive algebraic group and K a maximal compact subgroup of G. In [4], Casselman obtained the explicit formula of zonal spherical functions on G with respect to K on the assumption that K is special. It is known (Bruhat and Tits [3]) that the affine root system of algebraic group which has good but not special maximal compact subgroup is A1 C2, or Bn (n > 3), and all Bn-types can be realized by orthogonal groups. Here the assumption “good” is necessary for the Satake’s theory of spherical functions.

Theory of spherical functions on reductive algebraic groups over p-adic fields

Theory of spherical functions on reductive algebraic groups over p-adic fields

A Theory of Spherical Functions. I

A Theory of Spherical Functions. I

A theory of spherical functions. I

A theory of spherical functions. I