Abstract
We consider the problem of one-sided weighted integral approximation on the interval [−1, 1] to the characteristic functions of intervals (a, 1] ⊂ (−1, 1] and (a, b) ⊂ (−1, 1) by algebraic polynomials. In the case of half-intervals, the problem is solved completely. We construct an example to illustrate the difficulties arising in the case of an open interval.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. G. Babenko, Yu. V. Kryakin, and V. A. Yudin, “One-sided approximation in l of the characteristic function of an interval by trigonometric polynomials,” Proc. Steklov Inst. Math. 280 (Suppl. 1), S39–S52 (2013).
I. S. Berezin and N. P. Zhidkov, Computing Methods (Fizmatgiz, Moscow, 1962; Pergamon, Oxford, 1965), Vols. 1, 2.
V. I. Krylov, Approximate Calculation of Integrals (Fizmatgiz, Moscow, 1959) [in Russian].
A. A. Markov, Selected Works on the Theory of Continued Fractions and the Theory of Functions Least Deviating from Zero (Gostechizdat, Moscow, 1948) [in Russian].
A. G. Postnikov, Introduction to Analytic Number Theory (Nauka, Moscow, 1971; Amer. Math. Soc., New York, 1988).
G. Szegő, Orthogonal Polynomials (AMS, New York, 1959; Fizmatgiz, Moscow, 1962).
R. Bojanić and R. DeVore, “On polynomials of best one-sided approximation,” Enseign. Math. 2 (12), 139–164 (1966).
J. Bustamante, R. Martínez Cruz, and J. M. Quesada, “Quasi orthogonal Jacobi polynomials and best one-sided L1 approximation to step functions,” J. Approx. Theory 198, 10–23 (2015).
A. Bultheel, R. Cruz-Barroso, and M. Van Barel, “On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line,” Calcolo 47 (1), 21–48 (2010).
B. Beckermann, J. Bustamante, R. Martinez-Cruz, and J. M. Quesada, “Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa,” Calcolo 51 (2), 319–328 (2014).
X.-J. Li and J. D. Vaarler, “Some trigonometric extremal functions and the Erdös–Turán type inequalities,” Indiana Univ. Math. J. 48 (1), 183–236 (1999).
F. Peherstorfer, “Positive quadrature formulas. III: Asymptotics of weights,” Math. Comput. 77 (264), 2241–2259 (2008).
T. J. Stieltjes, “Quelques recherches sur la théorie des quadratures dites méchaniques,” Ann. Sci. Ecole Norm. Sup., Ser. 3, No. 1, 409–426 (1884).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.G. Babenko, M.V. Deikalova, Sz.G. Revesz, 2015, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2015, Vol. 21, No. 4.
Rights and permissions
About this article
Cite this article
Babenko, A.G., Deikalova, M.V. & Revesz, S.G. One-sided weighted integral approximation of characteristic functions of intervals by polynomials on a closed interval. Proc. Steklov Inst. Math. 297 (Suppl 1), 11–18 (2017). https://doi.org/10.1134/S0081543817050029
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543817050029