A local version of the Pawłucki–Pleśniak extension operator

Altun, Muhammed; Goncharov, Alexander

doi:10.1016/j.jat.2004.10.013

Cited by 7 publications

(10 citation statements)

References 18 publications

(28 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here we consider rather small Cantor-type sets that are neither Markov no local Markov. We follow [2] in our construction, so W is a local version of the Paw lucki-Pleśniak operator. It is interesting that, at least for small sets, W can be considered as an operator extending basis elements of the space.…”

Section: Three Methods Of Extensionmentioning

confidence: 99%

“…where Ω N and u are taken as in Lemma 5.2. We aim to show that (15) N s |A Since divided differences are symmetric in their arguments, we can use (4) from [2] :…”

Section: Extension Property Of Weakly Equilibrium Cantor-type Setsmentioning

confidence: 99%

“…For simplicity, we fix k j = j 2 that satisfies (19). Here, δ (2) k > k −2k ε 1 ε 2 · · · ε j for k j ≤ k < k j+1 . We aim to show that M n (K 2 )/M n (K 1 ) → 0 as n → ∞.…”

Section: Extension Property and Growth Of Markov's Factorsmentioning

confidence: 99%

See 2 more Smart Citations

Mityagin's extension problem. Progress report

Goncharov¹,

Ural²

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

Abstract. Given a compact set K ⊂ R d , let E(K) denote the space of Whitney jets on K. The compact set K is said to have the extension property if there exists a continuous linear extension operator W :In 1961 B. S. Mityagin posed a problem to give a characterization of the extension property in geometric terms. We show that there is no such complete description in terms of densities of Hausdorff contents or related characteristics. Also the extension property cannot be characterized in terms of growth of Markov's factors for the set.

show abstract

Section: Three Methods Of Extensionmentioning

confidence: 99%

“…where Ω N and u are taken as in Lemma 5.2. We aim to show that (15) N s |A Since divided differences are symmetric in their arguments, we can use (4) from [2] :…”

Section: Extension Property Of Weakly Equilibrium Cantor-type Setsmentioning

confidence: 99%

See 1 more Smart Citation

Mityagin's extension problem. Progress report

Goncharov¹,

Ural²

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…The crucial observation is that, under the Markov inequality assumption, the natural but rather awkward quotient topology is equivalent to a much simpler one, the so-called Jackson topology. Since then, various authors have proposed other such extension operators [31,1]. The discrete least squares method leads to another example.…”

Section: Theorem 9 If K Satisfies a Markov Inequality With Exponent mentioning

confidence: 99%

Uniform approximation by discrete least squares polynomials

Calvi

Levenberg²

2008

Journal of Approximation Theory

150

View full text Add to dashboard Cite

We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these polynomials provide nearly optimal approximants. For analytic functions, similar results may be achieved on more general K by allowing the number of points to grow at a slightly larger rate.

show abstract

“…As a method we employ local interpolations of functions that were used in [1] to present extension operators for the Whitney spaces E(K (α) ) and in [11] to construct topological bases in spaces E(K) for more general Cantor-type sets.…”

mentioning

confidence: 99%