Approximation by Piecewise Constant Functions on a Square

Kochurov, A. S.

doi:10.1023/b:matn.0000023336.90664.15

Cited by 3 publications

(5 citation statements)

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“…as N → ∞, then the Hesse matrix of function f is neither positive definite nor negative definite on Ω. We remark that similar results in the case d = 2 were obtained by a different method in [4]. Recently in [5] it was established that for 1 ≤ p, q ≤ ∞ which satisfy inequality…”

Section: Lower Estimates On the Saturation Order Of Approximationsupporting

confidence: 58%

Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

Kozynenko

2018

Vìsn. Dnìpro unìv., Ser. Mat.

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Section: Lower Estimates On the Saturation Order Of Approximationsupporting

confidence: 58%

Lower estimates on the saturation order of approximation of twice continuously differentiable functions by piecewise constants on convex partitions

Kozynenko

2018

Vìsn. Dnìpro unìv., Ser. Mat.

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“…If this condition is not satisfied, then we say that the partitions in the sequence are "anisotropic". It was shown in [4,5] (and earlier in [9] for d = 2) that on a wider set of all convex partitions D N significantly better order of approximation can be achieved. More precisely, for 1 p ∞,…”

Section: Introductionmentioning

confidence: 93%

“…Note that to obtain (9) in the case α = γ, instead of the Hölder inequality we only need to use non-negativity and subadditivity of function Φ:…”

Section: Cell Countingmentioning

confidence: 99%

Optimal approximation order of piecewise constants on convex partitions

Davydov

Kozynenko

Skorokhodov

2020

Journal of Complexity

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We prove that the error of the best nonlinear L p -approximation by piecewise constants on convex partitions is O N − 2 d+1 , where N is the number of cells, for all functions in the Sobolev space W 2 q (Ω) on a cube Ω ⊂ R d , d 2, as soon asis achieved on a polyhedral partition obtained by anisotropic refinement of an adaptive dyadic partition. Further estimates of the approximation order from the above and below are given for various Sobolev and Sobolev-Slobodeckij spaces W r q (Ω) embedded in L p (Ω), some of which also improve the standard estimate O N − 1 d known to be optimal on isotropic partitions.

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“…The order of piecewise constant approximation on anisotropic partitions in two dimensions has been investigated in [19]. It is shown that for any…”

Section: Theorem 10 ([10]mentioning

confidence: 99%

“…Note that by triangulating each polygonal cell of ∆ N one obtains a convex partition with O(N) triangular cells, so that Theorem 10 also applies, giving the same saturation order N −2/3 . Another result of [19] is that for any f ∈ C This algorithm is illustrated in Fig. 3.…”

Section: Theorem 10 ([10]mentioning

confidence: 99%