Series expansions among weighted Lebesgue function spaces and applications to positive definite functions on compact two-point homogeneous spaces

Abstract

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces Lw 1 ([−1, 1]), where the weight w(x) = wα,β(x) = (1 − x)α(1 + x)β is the one related to the Jacobi orthogonal polynomials P(α,β)(x) in [−1, 1], and the exponents α and β are related to the dimension d. We derive some recurrence relations among the coefficients of the series representations under different exponents, and we apply them to prove inheritance of positive definiteness between dimensions. Additionally, we give bounds on the curvature at the origin of such positive definite functions with compact support, extending the existing solutions from d-dimensional spheres to compact two-point homogeneous spaces.