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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)

- Frank Filbir, Hrushikesh Narhar Mhaskar
- J. Complexity
- 2011

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)

- Wolfgang zu Castell, Frank Filbir
- Journal of Approximation Theory
- 2005

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)

- F. FILBIR
- 2006

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)

- Wolfgang zu Castell, Frank Filbir, Ryszard Szwarc
- Journal of Approximation Theory
- 2005

- Frank Filbir, Hrushikesh Narhar Mhaskar, Jürgen Prestin
- Journal of Approximation Theory
- 2009

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

- Frank Filbir, Dominik Schmid
- Journal of Approximation Theory
- 2008

- Martin Ehler, Frank Filbir, Hrushikesh Narhar Mhaskar
- Journal of Computational Biology
- 2012

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)

- Frank Filbir, Stefan Kunis, Ruben Seyfried
- SIAM J. Numerical Analysis
- 2014

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)

- Frank Filbir, Daniel Potts
- 2009

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)