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

We investigate some spectral properties of a second order differential-difference operator J_α ,β on L^2((-π ,π ),dμ _α , β) , α ,β∈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 α ,β is ≤ -1 , we construct exotic orthonormal bases that consist of eigenfunctions of J_α ,β . This allows one to consider natural self-adjoint exotic extensions of J_α ,β and the corresponding exotic Jacobi–Dunkl and Jacobi–Dunkl–Poisson semigroups.