A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators
- Autores: Vinogradov O.1, Gladkaya A.1
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Afiliações:
- St.Petersburg State University
- Edição: Volume 217, Nº 1 (2016)
- Páginas: 3-22
- Seção: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/237956
- DOI: https://doi.org/10.1007/s10958-016-2950-7
- ID: 237956
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Resumo
Let σ > 0, m, r ∈ ℕ, m ≥ r, let Sσ,m be the space of splines of order m and minimal defect with nodes \( \frac{j\pi }{\sigma } \) (j ∈ ℤ), and let Aσ,m(f)p be the best approximation of a function f by the set Sσ,m in the space Lp(ℝ). It is known that for p = 1,+∞,
\( \begin{array}{l} \sup \hfill \\ {}f\in {W}_p^{(r)}\left(\mathbb{R}\right)\hfill \end{array}\frac{A_{\sigma, m}{(f)}_p}{{\left\Vert {f}^{(r)}\right\Vert}_p}=\frac{K_r}{\sigma^r}, \)
where Kr are the Favard constants. In this paper, linear operators Xσ,r,m with values in Sσ,m such that for all p ∈ [1,+∞] and f ∈ Wp(r)(ℝ),\( {\left\Vert f-{X}_{\sigma, r,m}(f)\right\Vert}_p\le \frac{K_r}{\sigma^r}{\left\Vert {f}^{(r)}\right\Vert}_p \)
are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.Sobre autores
O. Vinogradov
St.Petersburg State University
Autor responsável pela correspondência
Email: olvin@math.spbu.ru
Rússia, St.Petersburg
A. Gladkaya
St.Petersburg State University
Autor responsável pela correspondência
Email: anna.v.gladkaya@gmail.com
Rússia, St.Petersburg