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

Abstract

Let σ > 0, m, r ∈ ℕ, m ≥ r, 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)(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.