An error bound for the approximation of the curvature of graphs of periodic functions from the class $W^r$ for $r\ge 3$ in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions $W^r$ on a period or a closed interval for $r\ge 2$. There are several papers on curvature approximation for planar curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the $L^2$ norm), interpolation splines with uniform knots, Fej\'er means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the $C^{\infty}$ norm). The technique of this paper, based on the lemma, can possibly be generalized to the $L^p$ metric and other approximation methods.
The paper was received by the Editorial Office on June, 1, 2017.
Yurii Nikolaevich Subbotin, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: yunsub@imm.uran.ru
Nikolai Ivanovich Chernykh, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: chernykh@imm.uran.ru
REFERENCES
1. Subbotin Yu.N. Approximation of the curvature for certain smooth classes of plane curves by elements of finite-dimensional spaces. Izv. Tul. Gos. Univ. Estestvennye nauki. 2012, no. 3, pp. 41–47 (in Russian).
2. Subbotin Yu.N., Chernykh N.I. Interpolyatsionnye vspleski v zadache otsenki krivizny [Interpolation wavelets in the problem of estimating the curvature]. Proc. Internat. Summer Math. Stechkin School-Conf. on Function Theory. Dushanbe, Offset Publ., 2016, pp. 231–233.
3. Sokolov I.G. The remainder term of the Fourier series of differentiable functions. Dokl. Akad. Nauk SSSR, 1955, vol. 103, no. 1, pp. 23–26 (in Russian).
4. Achieser N.I. Theory of approximation. Reprint of the 1956, New York: Dover Publ., Inc., 1992, 307 p. ISBN: 0486671291 . Original Russian text published in Akhiezer N.I. Lektsii po teorii approksimatsii. Moscow, Leningrad: OGIZ Publ., 1947, 323 p.
