Harmonic Analysis Associated with the Jacobi–Dunkl Operator on (-pi ,pi ): Exotic Cases

Abstract

We investigate some spectral properties of a second order differential-difference operator J_{alpha ,beta } on L^2((-pi ,pi ),dmu _{alpha , beta }), alpha ,beta in mathbb {R}, called the Jacobi–Dunkl operator of compact type. Using an idea of Hajmirzaahmad, in exotic cases, e.g. when at least one of the two parameters alpha ,beta is le -1, we construct exotic orthonormal bases that consist of eigenfunctions of J_{alpha ,beta }. This allows one to consider natural self-adjoint exotic extensions of J_{alpha ,beta } and the corresponding exotic Jacobi–Dunkl and Jacobi–Dunkl–Poisson semigroups.