Hrushikesh Narhar Mhaskar

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Geodetic and meteorological data, collected via satellites for example, are genuinely scattered, and not confined to any special set of points. Even so, known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites (points) used or, more significantly, on the number of sites required. Here,(More)
We obtain a characterization of local Besov spaces of functions on [−1, 1] in terms of algebraic polynomial operators. These operators are constructed using the coefficients in the orthogonal polynomial expansions of the functions involved. The example of Jacobi polynomials is studied in further detail. A by-product of our proofs is an apparently simple(More)
A zonal function (ZF) network is a function of the form x 7→ ∑n k=1 ckφ(x · yk), where x and the yk’s are on the on the unit sphere in q+1 dimensional Euclidean space, and where the yk’s are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this degree of approximation with that obtained with the(More)
Let q ≥ 1 be an integer, Sq be the unit sphere embedded in the Euclidean space Rq+1. A Zonal Function (ZF) network with an activation function φ : [−1, 1] → R and n neurons is a function on Sq of the form x 7→ nk=1 akφ(x · ξk), where ak’s are real numbers, ξk’s are points on Sq. We consider the activation functions φ for which the coefficients {φ̂(`)} in(More)
The problem is said to be tractable if there exist constants c, α, β independent of q (but possibly dependent on μ and F) such that En(F , μ) ≤ cqαn−β. We explore different regions (including manifolds), function classes, and measures for which this problem is tractable. Our results include tractability theorems for integration with respect to non-tensor(More)
Let s > or = 1 be an integer. A Gaussian network is a function on Rs of the form [Formula: see text]. The minimal separation among the centers, defined by (1/2) min(1 < or = j not = k < or = N) [Formula: see text], is an important characteristic of the network that determines the stability of interpolation by Gaussian networks, the degree of approximation(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)