PIECEWISE-POLYNOMIAL APPROXIMATIONS OF FUNCTIONS OF THE CLASSES $ W_{p}^{\alpha}$

Бирман, М. Ш.; Solomjak, M Z

doi:10.1070/sm1967v002n03abeh002343

Cited by 211 publications

(201 citation statements)

References 2 publications

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“…We remark that in the situation of this proof, where we are concerned with the special case p = 2, i.e., approximation in W α,2 ([0, 1] 2 ) for α ∈ (1, 2), we obtain error bounds of the form (4.1) for non-adaptive quadtree partitions Q N of [0, 1] 2 (see [4]), consisting of quadtree elements, each of whose diameter is bounded above by C/ √ N , for some constant C > 0 independent of N (cf. Figure 5 for illustration).…”

Section: 2mentioning

confidence: 99%

“…In particular, an explicit algorithmic construction of a quadtree partition Q N satisfying the error estimate (4.1) is provided in [4]. But Q N does not necessarily minimize the approximation error in (4.1) among all quadtree partitions Q of size |Q| ≤ N .…”

Section: 1mentioning

confidence: 99%

“…To detail our analysis, we recall a classical result by Birman-Solomjak [4] concerning the approximation of regular functions by piecewise-affine functions over quadtree partitions of [0, 1] 2 . Before doing so, we first give a formal definition for quadtree partitions.…”

Section: 1mentioning

confidence: 99%

“…In Section 3, we show that linear splines over locally adaptive Delaunay triangulations lead to asymptotic N -term estimates of the form (1.4) for piecewise linear-affine horizon functions with α Hölder smooth boundary. Then, in Section 4 we first discuss a classical result by Birman-Solomjak [4], which states that piecewise affine-linear functions over quadtree partitions provide optimal approximation rates of the form (1.4) for regular functions f ∈ W α,p ([0, 1] 2 ), where α > 2/p − 1. We transfer the Birman-Solomjak result to the special case of linear spline approximation over locally adaptive Delaunay triangulations.…”

Section: Introductionmentioning

confidence: 99%

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Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Demaret

Iske

2014

Math. Comp.

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Abstract. Anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data, in particular from image data. In previous work, we have proposed a locally adaptive method for efficient image approximation, called adaptive thinning, which relies on linear splines over anisotropic Delaunay triangulations. In this paper, we prove asymptotically optimal N -term approximation rates for linear splines over anisotropic Delaunay triangulations, where our analysis applies to relevant classes of target functions: (a) piecewise linear horizon functions across α Hölder smooth boundaries, (b) functions of W α,p regularity, where α > 2/p−1, (c) piecewise regular horizon functions of W α,2 regularity, where α > 1.

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Section: 2mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Demaret

Iske

2014

Math. Comp.

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“…It follows using interpolation theory from the result of Birman and Solomyak [4]. However, the proof presented there is rather long and difficult.…”

Section: Introductionmentioning

confidence: 99%

Replicant compression coding in Besov spaces

Kerkyacharian¹,

Picard²

2003

ESAIM: PS

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Abstract. We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B s π,q on a regular domain ofThis proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication.Mathematics Subject Classification. 41A25, 41A46, 65F99, 65N12, 65N55.

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Some s‐numbers of embeddings of function spaces with weights of logarithmic type

Gąsiorowska

Skrzypczak

2012

Mathematische Nachrichten

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degenerate pseudodifferential operators, negative spectrum MSC (2010) 46E35, 42B35, 35P15, 47B06 Dedicated to Hans Triebel on the occasion of his 75th birthdayIn this paper we deal with compact embeddings of weighted function spaces of Besov and Triebel-Lizorkin type with weights of logarithmic growth near infinity. We obtain the exact estimates for the asymptotic behaviour of Gelfand, Kolmogorov and Weyl numbers of the embeddings. As an application we get eigenvalue estimates of some degenerate pseudodifferential operators of type. Finally we deal with the "negative spectrum" of operators Hγ = A − γV for γ → ∞, where the potential V is a positive function and A is a positive elliptic pseudodifferential operator. We concentrate on the case when the Weyl numbers and Pietsch's inequality give us better estimates then entropy numbers and Carl's inequality.

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PIECEWISE-POLYNOMIAL APPROXIMATIONS OF FUNCTIONS OF THE CLASSES $ W_{p}^{\alpha}$

Cited by 211 publications

References 2 publications

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Optimal $N$-term approximation by linear splines over anisotropic Delaunay triangulations

Replicant compression coding in Besov spaces

Some s‐numbers of embeddings of function spaces with weights of logarithmic type

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