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Analog of the Akhiezer–Krein–Favard Sums for Periodic Splines of Minimal Defect

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Abstract

Let \(n,r,m \in {\mathbb{N}},m \geqslant W_p^{\left( r \right)}\) be the space of 2π-periodic functions whose (r – 1)th-order derivative is absolutely continuous on any segment and rth-order derivative belongs to L p, S 2n,m is the space of 2π-periodic splines of order m of minimal defect over the uniform partition \(\frac{{k{\pi }}}{n}\left( {k \in {\mathbb{Z}}} \right)\). In this paper, we construct linear operators \(X_{n,r,m} :L_1 \to S_{2n,m}\) such that

$$\mathop {\sup }\limits_{f \in W_\infty ^{\left( r \right)} } \frac{{\left\| {f - X_{n,r,m} \left( f \right)} \right\|_\infty4}}{{\left\| {f^{\left( r \right)} } \right\|_\infty }} = \mathop {\sup }\limits_{f \in W_1^{\left( r \right)} } \frac{{\left\| {f - X_{n,r,m} \left( f \right)} \right\|_1 }}{{\left\| {f^{\left( r \right)} } \right\|_1 }} = \frac{{K_r }}{{n^r }},$$

where

$$K_r = \frac{4}{{\pi }}\sum\limits_{l = 0}^\infty {\frac{{\left( { - 1} \right)^{l\left( {r + 1} \right)} }}{{\left( {2l + 1} \right)^{r + 1} }}.}$$

To construct the operators X n,r,m, we use the same idea as in the polynomial case, i.e., the interpolation of Bernoulli kernels. As is proved, the operators X n,r,m converge to polynomial Akhiezer–Krein–Favard operators as \(m \to \infty\). Bibliography: 10 titles.

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Vinogradov, O.L. Analog of the Akhiezer–Krein–Favard Sums for Periodic Splines of Minimal Defect. Journal of Mathematical Sciences 114, 1608–1627 (2003). https://doi.org/10.1023/A:1022360711364

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