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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)
A zonal function (ZF) network is a function of the form x → n k=1 c k φ(x · y k), where x and the y k 's are on the on the unit sphere in q + 1 dimensional Euclidean space, and where the y k '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(More)
Let q ≥ 1 be an integer, Q be a Borel subset of the Euclidean space R q , µ be a probability measure on Q, and F be a class of real valued, µ-integrable functions on Q. The complexity problem of approximating fdµ using quasi-Monte Carlo methods is to estimate sup f ∈F fdµ − 1 n n k=1 f (x k) The problem is said to be tractable if there exist constants c, α,(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)
For the unit sphere embedded in a Euclidean space, we obtain quadrature formulas that are exact for spherical harmonics of a fixed order, have nonnegative weights, and are based on function values at scattered points (sites). The number of scattered sites required is comparable to the dimension of the space for which the quadrature formula is required to be(More)