Sharp asymptotics of the L p approximation error for interpolation on block partitions

Abstract

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in R d . We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.

Error estimates for harmonic and biharmonic interpolation splines with annular geometry

“…the discussion in [27]. Harmonic splines for block partitions have been discussed by various authors in the literature, see [5], [6], [7], [29]. Here a set Ω of the form [a…”

“…The existence of a harmonic spline interpolant is easy to establish when we assume that the Dirichlet problem is solvable 1 for each domain Ω j for j = 1, ..., N. The uniqueness of the harmonic interpolant is a simple consequence of the maximum principle. The main results in [7], [5], [6], [29] are optimal estimates for the error E (F ) := F − I 2 (F ) for a given twice differentiable function F ∈ C 2 Ω in the case of block partitions with respect to the supremum norm defined by…”

On theLp-error of adaptive approximation of bivariate functions by harmonic splines

“…In this paper we shall discuss harmonic spline "interpolation" (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the L p -error for adaptive approximation by harmonic splines.Keywords interpolation · adaptive · harmonic spline · optimal error · asymptotics Mathematics Subject Classification (2000) 41A15 · 41A60 · 68W25 · 97N50 1 Harmonic splines as an alternative to polynomial splines In [5] authors investigated the question of adaptive approximation by polynomial splines on box partitions in R d . The obtained a general description as well as sharp constant in several popular cases.…”

“…Keywords interpolation · adaptive · harmonic spline · optimal error · asymptotics Mathematics Subject Classification (2000) 41A15 · 41A60 · 68W25 · 97N50 1 Harmonic splines as an alternative to polynomial splines In [5] authors investigated the question of adaptive approximation by polynomial splines on box partitions in R d . The obtained a general description as well as sharp constant in several popular cases.…”

“…1 Harmonic splines as an alternative to polynomial splines In [5] authors investigated the question of adaptive approximation by polynomial splines on box partitions in R d . The obtained a general description as well as sharp constant in several popular cases.…”