Journal of Mathematical Sciences

, Volume 217, Issue 1, pp 3–22 | Cite as

A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators

  • O. L. Vinogradov
  • A. V. Gladkaya
Let σ > 0, m, r ∈ ℕ, mr, let S σ,m be the space of splines of order m and minimal defect with nodes \( \frac{j\pi }{\sigma } \) (j ∈ ℤ), and let A σ,m (f) p be the best approximation of a function f by the set S σ,m in the space L p (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 K r are the Favard constants. In this paper, linear operators X σ,r,m with values in S σ,m such that for all p ∈ [1,+∞] and f ∈ W p (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.


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.St.Petersburg State UniversitySt.PetersburgRussia

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