Best methods of interpolation for certain classes of differentiable functions

Boyanov, B D

doi:10.1007/bf01105379

Cited by 12 publications

(4 citation statements)

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“…But if sup r(H(y)) < oo, yEIK the only case of interest, is a c 0 (y)EZ such that llux-c 6 (y) II ~ for any xEK with Ix = y. [3]). In general c(y) is a nonlinear function of y.…”

mentioning

confidence: 99%

A Survey of Optimal Recovery

Micchelli

Rivlin

1977

Optimal Estimation in Approximation Theory

257

113

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“…But if sup r(H(y)) < oo, yEIK the only case of interest, is a c 0 (y)EZ such that llux-c 6 (y) II ~ for any xEK with Ix = y. [3]). In general c(y) is a nonlinear function of y.…”

mentioning

confidence: 99%

A Survey of Optimal Recovery

Micchelli

Rivlin

1977

Optimal Estimation in Approximation Theory

257

113

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“…The problem of optimal recovery of functions from their values and values of their derivatives of order up to r − 1 at n nodes was considered by Bojanov [4] on the class of functions defined on an interval whose (r − 1)-th derivative is absolutely continuous and whose r-th derivative is bounded in L p -norm. He proved optimality of equally spaced nodes.…”

Section: Problemmentioning

confidence: 99%

Optimal recovery of isotropic classes of twice-differentiable multivariate functions

Бабенко

Borodachov

Skorokhodov

2010

Journal of Complexity

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a b s t r a c tWe consider the class of functions defined on a convex body in R d , d ∈ N, whose second derivatives in any direction are uniformly bounded and the class of d-variate functions periodic with respect to a given full-rank lattice L and having uniformly bounded second derivative in any direction. The problem of the optimal algorithm which recovers functions from these classes using their values and values of their gradients at n points (nodes) is considered. We first obtain an estimate for the error of the optimal algorithms with fixed nodes. In the periodic case, for every n sufficiently large, we describe the optimal set of n nodes. When d = 2, for certain periodic cases, optimality of the hexagonal arrangement of nodes is shown. For both the periodic case and the non-periodic case we present asymptotic results as n gets large.Published by Elsevier Inc.

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“…Some ideas used in this proof appear in [5]. By the Taylor's formula, f (v) can be written as follows…”

Section: Theorem 41 Let T Be a Non-degeneratementioning

confidence: 99%

“…We show that the optimal method of recovery is given by a quadratic interpolating spline over a Delaunay triangulation of X in R d . We note that for d = 1, this problem was solved in [5] in a more general setting. However, our optimal recovery spline method is new even in the univariate case and, in particular, is different from the one obtained in [5].…”

Section: Introductionmentioning

confidence: 97%

An optimal multivariate spline method of recovery of twice differentiable functions

Borodachov

Sorokina

2011

Bit Numer Math

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A continuous quadratic polynomial spline of several variables is constructed. It solves the optimal recovery problem studied by V.F. Babenko, S.V. Borodachov, and D.S. Skorokhodov for the class of functions defined on a convex polytope in R d , whose second derivatives in any direction are uniformly bounded, and for a periodic analogue of this class. The information consists of the values and gradients of the function at some finite set of nodes in R d .

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Best methods of interpolation for certain classes of differentiable functions

Cited by 12 publications

References 0 publications

A Survey of Optimal Recovery

A Survey of Optimal Recovery

Optimal recovery of isotropic classes of twice-differentiable multivariate functions

An optimal multivariate spline method of recovery of twice differentiable functions

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