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

摘要: Let σ > 0, m, r ∈ ℕ, m ≥ r, let S σ,m be the space of splines order and minimal defect with nodes $$ \frac{j\pi }{\sigma } $$ (j ℤ), A (f) p best approximation a function f by set in L (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 Favard constants. In this paper, linear operators X σ,r,m values such all [1,+∞] f ∈ W () (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 upper bounds indicated above can achieved methods approximation, which was previously unknown. Bibliography: 21 titles.