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

V. S. Barbosa, P. Gregori, A. P. Peron, E. Porcu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Geodesically isotropic positive definite functions on compact two-point homogeneous spaces of dimension d have series representation as members of weighted Lebesgue spaces L1w([−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.

Original languageBritish English
Article number126487
JournalJournal of Mathematical Analysis and Applications
Volume516
Issue number1
DOIs
StatePublished - 1 Dec 2022

Keywords

  • Compact two-point homogeneous spaces
  • Curvature at the origin
  • Jacobi polynomials
  • Positive definite functions
  • Recurrence relations

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