Frank Filbir

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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(−`2kt)φk(x)φk(y). If T is a (differential) operator, and both Kt and TyKt have(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.(More)
Since radial positive definite functions on R 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 the(More)
Starting from a natural generalization of the trigonometric case, we construct a de la Vallée Poussin approximation process in the uniform and L norms. With respect to the classical approach we obtain the convergence for a wider class of Jacobi weights. Even if we only consider the Jacobi case, our construction is very general and can be extended to other(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)
where Sd−1 = { u ∈ Rd : |u| = 1 } denotes the unit sphere and σ its surface measure. In particular is σ(S1) = 2π and σ(S2) = 4π. The variable t ≥ 0 is called measurement time and the variable ξ ∈ Rd detector position or center point. In all practical applications these center points are located on a curve or surface and we consider the classical case ξ ∈(More)
In this technical report we present a complete proof of the Marcinkiewicz-Zygmund inequality on the bi-Sphere from [1, Theorem 4.5]. We use the same notation as in [1]. The Theorem 4.5 of [1] reads as follows: Theorem 1. Let N ∈ N2 and R be a admissible decomposition of S2 × S2 according to a sampling set X such that ‖R‖ ≤ η 21 Ñ , (1) where η ∈ (0, 1) is(More)