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 $$J_{\alpha ,\beta }$$ J α , β on $$L^2((-\pi ,\pi ),d\mu _{\alpha , \beta })$$ L 2 ( - π ) d μ $$\alpha \in \mathbb {R}$$ ∈ R called the Jacobi–Dunkl compact type. Using an idea Hajmirzaahmad, in exotic cases, e.g. when at least one two parameters $$ is $$\le -1$$ ≤ 1 we construct orthonormal bases that consist eigenfunctions . This allows to consider natural self-adjoint extensions and corresponding Jacobi–Dunkl–Poisson semigroups.