We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabó operator have constant eigenvalues on their domains of definition. This provides new and non-trivial examples of Osserman, Szabó, and IP manifolds. We also study when the associated Jordan normal form of these operators is constant. Subject Classification: 53B20.

Based 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 jx1jã1 Ð Ð Ð jxdjãd 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.

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl and derivative is shown to have a realization in terms of oscillator operators one dimension. This observation rests on fact intertwining maps Hermite polynomials generalized polynomials.

Contents 1. Introduction 1 2. Geometrical setting near a dilated catenoid 6 3. Jacobi-Toda system on the Catenoid 9 4. Jacobi operator and the linear Jacobi-Toda operator on the catenoid. 16 5. Approximation of the solution of the theorem 1 23 6. Proof of theorem 1. 30 7. gluing reduction and solution to the projected problem.

In this note, a new proof for the positivity of Dunkl’s intertwining operator in the crystallographic case is given. It is based on an asymptotic relationship between the Opdam-Cherednik kernel and the Dunkl kernel as recently observed by M. de Jeu, and on positivity results of S. Sahi for the Heckman-Opdam polynomials and their nonsymmetric counterparts. 2000 AMS Subject Classification: 33C52,...

The goal of this paper is to present a Dunkl-Gamma type operator with the help generalization two-variable Hermite polynomials and derive its approximating properties via classical modulus continuity, second continuity Peetre’s K-functional.

We present a technique for reconstructing a semi-infinite Jacobi operator in the limit circle case from the spectra of two different self-adjoint extensions. Moreover, we give necessary and sufficient conditions for two real sequences to be the spectra of two different self-adjoint extensions of a Jacobi operator in the limit circle case. ∗Mathematics Subject Classification(2000): 47B36, 49N45,...

Let M be a compact (two-sided) minimal hypersurface in a Riemannian manifold M n+1 . It is a simple fact that if M has positive Ricci curvature then M cannot be stable (i. e. its Jacobi operator L has index at least one). If M = S(1) is the unit sphere and L has index one, then it is known that M must be a totally geodesic equator. We prove that if M is the real projective space RP = Sn+1(1)/{±...