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Let {φk} be an orthonormal system on a quasi-metric measure space X, {k} be a nondecreasing sequence of numbers with limk→∞ k = ∞. A diffusion polynomial of degree L is an element of the span of {φk : k ≤ L}. The heat kernel is defined formally by Kt(x, y) = P ∞ k=0 exp(− 2 k t)φk(x)φk(y). If T is a (differential) operator, and both Kt and TyKt have(More)
Let α, β ≥ −1/2, and for k = 0, 1, · · ·, pk (α,β) denote the orthonormalized Jacobi polynomial of degree k. We discuss the construction of a matrix H so that there exist positive constants c, c1, depending only on H, α, and β such that ˛ ˛ ˛ ˛ ˛ ∞ X k=0 Hk,npk (α,β) (cos θ)pk (α,β) (cos ϕ) ˛ ˛ ˛ ˛ ˛ ≤ c1n 2 max(α,β)+2 exp(−cn(θ − ϕ) Specializing to the(More)
Since radial positive definite functions on R d remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of(More)
Let X be a compact, connected, Riemannian manifold (without boundary), ρ be the geodesic distance on X, µ be a probability measure on X, and {φk} be an orthonormal (with respect to µ) system of continuous functions, φ0(x) = 1 for all x ∈ X, {k} ∞ k=0 be an nondecreasing sequence of real numbers with 0 = 1, k ↑ ∞ as k → ∞, ΠL := span {φj : j ≤ L}, L ≥ 0. We(More)
Diffusion geometry techniques are useful to classify patterns and visualize high-dimensional datasets. Building upon ideas from diffusion geometry, we outline our mathematical foundations for learning a function on high-dimension biomedical data in a local fashion from training data. Our approach is based on a localized summation kernel, and we verify the(More)
The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Recently, kernel methods for the(More)
We consider interpolation methods defined by positive definite functions on a compact group. Estimates for the smallest and largest eigenvalue of the interpolation matrix in terms of the localization of the positive definite function on G are presented and we provide a method to get positive definite functions explicitly on compact semisimple Lie groups.(More)