Abstract
Local parabolic splines on the axis \(\mathbb R\) with equidistant nodes realizing the simplest local approximation scheme are considered. But, instead of the values of functions at the nodes, its formula approximates their mean values in symmetric neighborhoods of these nodes. For an arbitrary averaging step \(H\) more than twice as large as the grid step \(h\) of the spline, the approximation errors are calculated exactly in the uniform metric of these functions and their derivatives for the function class \(W_\infty^2\). For small averaging steps \(H\le 2h\), these quantities were calculated by E. V. Strelkova (Shevaldina) in 2007.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. N. Subbotin, “Extremal problems of functional interpolation, and mean interpolation splines,” Proc. Steklov Inst. Math. 138, 127–185 (1977).
Yu. N. Subbotin, “Extremal functional interpolation in the mean with least value of the \(n\)th derivative for large averaging intervals,” Math. Notes 59 (1), 83–96 (1996).
Yu. N. Subbotin, “Extremal \(L_p\) interpolation in the mean with intersecting averaging intervals,” Izv. Math. 61 (1), 183–205 (1997).
A. A. Ligun and A. A. Shumeiko, Asymptotic Methods for the Recovery of Curves (Izd. Inst. Mat. NAN Ukr., Kiev, 1996) [in Russian].
V. T. Shevaldin, “Some problems of extremal interpolation in the mean for linear differential operators,” Proc. Steklov Inst. Math. 164, 233–273 (1985).
V. T. Shevaldin, “Extremal interpolation in the mean with overlapping averaging intervals and \(L\)-splines,” Izv. Math. 62 (4), 833–856 (1998).
E. V. Strelkova (Shevaldina), “Approximation by local parabolic splines of functions from their values in the mean,” Trudy Inst. Mat. i Mekh. UrO RAN 13 (4), 169–189 (2007).
H. Behforooz, “Approximation by integro cubic splines,” Appl. Math. Comput. 175 (1), 8–15 (2006).
H. Behforooz, “Interpolation by integro quintic splines,” Appl. Math. Comput. 216 (2), 364–367 (2010).
Yu. N. Subbotin, “Inheritance of monotonicity and convexity in local approximations,” Comput. Math. Math. Phys. 33 (7), 879–884 (1993).
N. P. Korneichuk, Splines in Approximation Theory (Nauka, Moscow, 1984) [in Russian].
V. T. Shevaldin, “Approximation by local parabolic splines with arbitrary knots,” Sib. Zh. Vychisl. Mat. 8 (1), 77–88 (2005).
Funding
This work is part of the research carried out at the Ural Mathematical Center.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Shevaldin, V.T. Local Approximation by Parabolic Splines in the Mean for Large Averaging Intervals. Math Notes 108, 733–742 (2020). https://doi.org/10.1134/S0001434620110127
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620110127