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Résumé |
Schoenberg’s theorem on positive definite functions on spheres manifests in three distinct “avatars”: in the study of matrix preservers, the harmonic analysis of homogeneous spaces, and the representation theory of infinite-dimensional groups. I will discuss these connections, starting from the classical entrywise calculus of the Schur product theorem and extending to the asymptotic behavior of spherical functions on Olshanski spherical pairs.
As a geometric application, we will see how this framework recovers the linear programming bounds of Delsarte–Goethals–Seidel and Kabatiansky–Levenshtein for spherical codes, which in turn yield upper bounds for sphere packing densities. I will then extend Schoenberg’s classical framework to partially defined positivity preservers on discrete domains, and apply this machinery to the problem of soft thresholding for correlation matrices in high-dimensional statistics.
This is joint work with James Pascoe. |
