On analogue of one problem of Erdös for non-periodic splines on the real domain

V.A. Kofanov

doi:10.15421/241315

On analogue of one problem of Erdös for non-periodic splines on the real domain

V.A. Kofanov (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0003-0392-2257

Abstract

We solve the analog of some problem of Erdös about the characterization of the non-periodic spline of order r and of minimal defect, with knots at the points $$$kh$$$, $$$k\in \mathbb{Z}$$$ and fixed uniform norm that has maximal arc lens over any fixed interval.

Keywords

inequalities for splines; problem of Erdös

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References

Pinkus A., Shisha O. "Variations on the Chebyshev and $$$L^q$$$-Theories of Best Approximation", Journal of Approx. Theory, 1982; 35(2): pp. 148-168. doi:10.1016/0021-9045(82)90033-8

Kofanov V.A. "Exact upper bounds of norms of functions and their derivatives on the classes of functions with given comparison function", Ukrainian Math. J., 2011; 63(7): pp. 969-984. (in Russian) doi:10.1007/s11253-011-0567-z

Bojanov B., Naidenov N. "An extension of the Landau-Kolmogorov inequality. Solution of a problem of Erdös", J. d'Analyse Mathematique, 1999; 78: pp. 263-280. doi:10.1007/BF02791137

Erdös P. "Open problems", Open Problems in Approximation Theory (B. Bojanov, ed.), SCT Publishing, Singapur, 1994; pp. 238-242.

Magaril-Il'yaev G.G. "Best approximations of functional classes by splines on the real domain", Trudy mat. in-ta im. Steklova, 1992; pp. 153-154. (in Russian)

Kofanov V.A. "Some extremum problems in different metrics for differentiable functions on the real domain", Ukrainian Math. J., 2009; 61(6): pp. 765-776. (in Russian) doi:10.1007/s11253-009-0254-5

Kolmogorov A.N. "On inequalities between upper bounds of consecutive derivatives of the function on the infinite interval", Izbr. tr.: Matematika, mechanika, Moscow, 2003; pp. 252-263. (in Russian)

Tikhomirov V.M. "Set widths in functional spaces and theory of the best approximations", Uspekhi mat. nauk, 1960; 15(3): pp. 81-120. (in Russian)