Orthogonal polynomial kernels and canonical correlations for Dirichlet measures

We consider a multivariate version of the so-called Lancaster problem of characterizing canonical correlation coefficients of symmetric bivariate distributions with identical marginals and orthogonal polynomial expansions. The marginal distributions examined in this paper are the Dirichlet and the Dirichlet multinomial distribution, respectively, on the continuous and the N-discrete d-dimensional simplex. Their infinite-dimensional limit distributions, respectively, the Poisson-Dirichlet distribution and Ewens's sampling formula, are considered as well. We study, in particular, the possibility of mapping canonical correlations on the d-dimensional continuous simplex (i) to canonical correlation sequences on the d+1-dimensional simplex and/or (ii) to canonical correlations on the discrete simplex, and vice versa. Driven by this motivation, the first half of the paper is devoted to providing a full characterization and probabilistic interpretation of n-orthogonal polynomial kernels (i.e., sums of products of orthogonal polynomials of the same degree n) with respect to the mentioned marginal distributions. We establish several identities and some integral representations which are multivariate extensions of important results known for the case d=2 since the 1970s. These results, along with a common interpretation of the mentioned kernels in terms of dependent Polya urns, are shown to be key features leading to several non-trivial solutions to Lancaster's problem, many of which can be extended naturally to the limit as $d\rightarrow\infty$.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ403 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Heat kernel generated frames in the setting of Dirichlet spaces

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on Rd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on Rd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincar\'e inequality which lead to heat kernels with small time Gaussian bounds and H\"older continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting

Genuinely sharp heat kernel estimates on compact rank-one symmetric spaces, for Jacobi expansions, on a ball and on a simplex

We prove genuinely sharp two-sided global estimates for heat kernels on all compact rank-one symmetric spaces. This generalizes the authors' recent result obtained for a Euclidean sphere of arbitrary dimension. Furthermore, similar heat kernel bounds are shown in the context of classical Jacobi expansions, on a ball and on a simplex. These results are more precise than the qualitatively sharp Gaussian estimates proved recently by several authors.Comment: 16 page

APS η\eta-invariant, path integrals, and mock modularity

We show that the Atiyah-Patodi-Singer $\eta$-invariant can be related to the temperature dependent Witten index of a noncompact theory and give a new proof of the APS theorem using scattering theory. We relate the $\eta$-invariant to a Callias index and compute it using localization of a supersymmetric path integral. We show that the $\eta$-invariant for the elliptic genus of a finite cigar is related to quantum modular forms obtained from the completion of a mock Jacobi form which we compute from the noncompact path integral.Comment: 44 pages, 5 figue