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
where
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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DOI: https://doi.org/10.1023/A:1022360711364