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,+∞,
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 (r) p (ℝ),
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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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 440, 2015, pp. 8–35.
Translated by O. L. Vinogradov and A. V. Gladkaya.
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Vinogradov, O.L., Gladkaya, A.V. A Nonperiodic Spline Analog of the Akhiezer–Krein–Favard Operators. J Math Sci 217, 3–22 (2016). https://doi.org/10.1007/s10958-016-2950-7
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DOI: https://doi.org/10.1007/s10958-016-2950-7