Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

1997 ◽  
Vol 49 (1) ◽  
pp. 175-192 ◽  
Author(s):  
Yuan Xu
Keyword(s):  
Orthogonal Polynomials ◽  
Spherical Harmonics ◽  
Unit Sphere ◽  
Poisson Kernel ◽  
Reflection Group ◽  
Weight Functions ◽  
Intertwining Operator ◽  
Finite Reflection Group ◽  
Explicit Formulae ◽  
Product Weight

AbstractBased on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions |x1|α1 . . . |xd|αd on the unit sphere Sd-1 in ℝd. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

scholarly journals Uncertainty Principles on Weighted Spheres, Balls, and Simplexes

2016 ◽  
Vol 59 (01) ◽  
pp. 62-72
Author(s):  
Han Feng
Keyword(s):  
Weight Function ◽  
Uncertainty Principle ◽  
Unit Sphere ◽  
Reflection Group ◽  
Beltrami Operator ◽  
Uncertainty Principles ◽  
Finite Reflection Group ◽  
Heisenberg Inequality ◽  
Full Analogy

Abstract This paper studies the uncertainty principle for spherical h-harmonic expansions on the unit sphere of ℝ d associated with a weight function invariant under a general finite reflection group, which is in full analogy with the classical Heisenberg inequality. Our proof is motivated by a new decomposition of the Dunkl–Laplace–Beltrami operator on the weighted sphere.

A New Composite Quadrature Rule

2013 ◽  
Vol 5 (04) ◽  
pp. 595-606
Author(s):  
Weiwei Sun ◽  
Qian Zhang
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Quadrature Rule ◽  
Numerical Results ◽  
Weight Functions

AbstractWe present a new composite quadrature rule which is exact for polynomials of degree 2N+K– 1 withNabscissas at each subinterval andKboundary conditions. The corresponding orthogonal polynomials are introduced and the analytic formulae for abscissas and weight functions are presented. Numerical results show that the new quadrature rule is more efficient, compared with classical ones.

Orthogonal Polynomials with Symmetry of Order Three

1984 ◽  
Vol 36 (4) ◽  
pp. 685-717 ◽  
Author(s):  
Charles F. Dunkl
Keyword(s):  
Orthogonal Polynomials ◽  
Unit Sphere ◽  
Jacobi Polynomials ◽  
Orthogonal Basis ◽  
Lower Degree ◽  
Surface Measure ◽  
Rotation Invariant ◽  
General Measure ◽  
Invariant Surface ◽  
Image Position

The measure (x1x2x3)2adm(x) on the unit sphere in R3 is invariant under sign-changes and permutations of the coordinates; here dm denotes the rotation-invariant surface measure. The more general measurecorresponds to the measureon the triangle(where ). Appell ([1] Chap. VI) constructed a basis of polynomials of degree n in v1, v2 orthogonal to all polynomials of lower degree, and a biorthogonal set for the case γ = 0. Later Fackerell and Littler [6] found a biorthogonal set for Appell's polynomials for γ ≠ 0. Meanwhile Pronol [10] had constructed an orthogonal basis in terms of Jacobi polynomials.

scholarly journals Twisted Invariant Theory for Reflection Groups

2006 ◽  
Vol 182 ◽  
pp. 135-170 ◽  
Author(s):  
C. Bonnafé ◽  
G. I. Lehrer ◽  
J. Michel
Keyword(s):  
Invariant Theory ◽  
Regular Element ◽  
Central Element ◽  
Reflection Group ◽  
General Criterion ◽  
Coordinate Ring ◽  
Reflection Groups ◽  
Complex Vector ◽  
Finite Reflection Group ◽  
Module Structures

AbstractLet G be a finite reflection group acting in a complex vector space V = ℂr, whose coordinate ring will be denoted by S. Any element γ ∈ GL(V) which normalises G acts on the ring SG of G-invariants. We attach invariants of the coset Gγ to this action, and show that if G′ is a parabolic subgroup of G, also normalised by γ, the invariants attaching to G′γ are essentially the same as those of Gγ. Four applications are given. First, we give a generalisation of a result of Springer-Stembridge which relates the module structures of the coinvariant algebras of G and G′ and secondly, we give a general criterion for an element of Gγ to be regular (in Springer’s sense) in invariant-theoretic terms, and use it to prove that up to a central element, all reflection cosets contain a regular element. Third, we prove the existence in any well-generated group, of analogues of Coxeter elements of the real reflection groups. Finally, we apply the analysis to quotients of G which are themselves reflection groups.

scholarly journals Distance to the convex hull of an orbit under the action of a compact Lie group

1999 ◽  
Vol 66 (3) ◽  
pp. 331-357 ◽  
Author(s):  
Randall R. Holmes ◽  
Tin-Yau Tam
Keyword(s):  
Convex Hull ◽  
Analogous Result ◽  
Maximal Compact Subgroup ◽  
Reductive Group ◽  
Compact Subgroup ◽  
Adjoint Action ◽  
Reflection Group ◽  
Compact Lie Group ◽  
Real Vector ◽  
Finite Reflection Group

AbstractFor a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

Complex Weight Functions for Classical Orthogonal Polynomials

1991 ◽  
Vol 43 (6) ◽  
pp. 1294-1308 ◽  
Author(s):  
Mourad E. H. Ismail ◽  
David R. Masson ◽  
Mizan Rahman
Keyword(s):  
Orthogonal Polynomials ◽  
Weight Functions ◽  
Classical Orthogonal Polynomials ◽  
The Real ◽  
Complex Functions ◽  
Wilson Polynomials ◽  
Distributional Weight

AbstractWe give complex weight functions with respect to which the Jacobi, Laguerre, little q-Jacobi and Askey-Wilson polynomials are orthogonal. The complex functions obtained are weight functions in a wider range of parameters than the real weight functions. They also provide an alternative to the recent distributional weight functions of Morton and Krall, and the more recent hyperfunction weight functions of Kim.

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