M.V. Deikalova, A.Yu. Torgashova. Best one-sided approximation in the mean of the characteristic function of an interval by algebraic polynomials ... P. 110-125

Let $\upsilon$ be a weight on $(-1,1)$, i.e., a measurable integrable nonnegative function different from zero almost everywhere on $(-1,1)$. Denote by $L^\upsilon(-1,1)$ the space of real-valued functions $f$ integrable with weight $\upsilon$ on $(-1,1)$ with the norm $\|f\|=\int_{-1}^{1}|f(x)|\upsilon(x)\,dx$. We consider the problems of the best one-sided approximation (from below and from above) in the space $L^\upsilon(-1,1)$ of the characteristic function of an interval $(a,b)$, $-1<a<b<1$, by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case when $a$ and $b$ are nodes of a positive quadrature formula under some conditions on its degree of precision as well as in the case of a symmetric interval $(-h,h),$ $0<h<1$, for an even weight $\upsilon$.

Keywords: one-sided approximation, characteristic function of an interval, algebraic polynomials

Received September 01, 2018

Revised October 09, 2018

Accepted October 15, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Marina Valer’evna Deikalova, Cand. Sci. (Phys.-Math.), Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: marina.deikalova@urfu.ru

Anastаsiya Yur’evna Torgashova, graduate student, Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: anastasiya.torgashova@mail.ru

REFERENCES

1.   Bojanic R., DeVore R. On polynomials of best one-sided approximation. Enseign. Math., 1966, vol. 2, no. 12, pp. 139–164.

2.   Krylov V.I. Approximate calculation of integrals. Mineola, N.Y.: Dover, 2006, 368 p. ISBN: 0486445798 . Original Russian text published in Krylov V.I. Priblizhennoe vychislenie integralov. Moscow: Fizmatlit Publ., 1959, 327 p.

3.   Korneichuk N.P., Ligun A.A., and Doronin V.G. Approksimatsiya s ogranicheniyami [Approximation with Constraints]. Kiev: Naukova Dumka, 1982, 254 p.

4.   Bustamante J., Quesada J.M., Martinez-Cruz R. Best one-sided $L_1$ approximation to the Heaviside and sign functions. J. Approx. Theory, 2012, vol. 164, pp. 791–802. doi: 10.1016/j.jat.2012.02.006

5.   Li X.-J., Vaaler J.D. Some trigonometric extremal functions and the Erd$\ddot{\mathrm{o}}$s–Tur$\acute{\mathrm{a}}$n type inequalities. Ind. Univ. Math. J., 1999, vol. 48, no. 1, pp. 183–236. doi: 10.1512/iumj.1999.48.1508

6.   Motornyi V.P., Motornaya O.V., Nitiema P.K. One-sided approximation of a step by algebraic polynomials in the mean. Ukr. Math. J., 2010, vol. 62, no. 3, pp. 467–482. doi: 10.1007/s11253-010-0366-y

7.   Bustamante J., Martinez-Cruz R., Quesada J.M. Quasi orthogonal Jacobi polynomials and best one-sided $L_1$ approximation to step functions. J. Approx. Theory, 2015, vol. 198, pp. 10–23. doi: 10.1016/j.jat.2015.05.001

8.   Babenko A.G., Kryakin Yu.V., and Yudin V.A. One-sided approximation in L of the characteristic function of an interval by trigonometric polynomials. Proc. Steklov Inst. Math., 2013, vol. 280, suppl. 1, pp. 39–52. doi: 10.1134/S0081543813020041

9.   Babenko A.G., Deikalova M.V., Revesz Sz.G. Weighted one-sided integral approximations to characteristic functions of intervals by polynomials on a closed interval. Proc. Steklov Inst. Math., 2017, vol. 297, suppl. 1, pp. 11–18. doi: 10.1134/S0081543817050029

10.   Beckermann B., Bustamante J., Martinez-Cruz R., Quesada J.M. Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa. Calcolo, 2014, vol. 51, no. 2, pp. 319–328. doi: 10.1007/s10092-013-0087-3

11.   Berezin I.S., Zhidkov N.P. Computing Methods, vol. 1. Oxford: Pergamon, 1965, 464 p. ISBN: 9781483180588 . Original Russian text published in Berezin I.S., Zhidkov N.P. Metody vychislenii, vol. 1. Moscow: Fizmatgiz Publ., 1962, 464 p.

12.   Arestov V.V., Berdyshev V.I., Chernykh N.I., Demina T.V., Kholschevnikova N.N., Konyagin S.V., Subbotin Yu.N., Telyakovskii S.A., Tsar’kov I.G., Yudin V.A. Exposition of the lectures by S.B. Stechkin on approximation theory. Eurasian Math. J., 2011, vol. 2, no. 4, pp. 5–155.

13.   Suetin P.K. Klassicheskie ortogonal’nye mnogochleny (Classical orthogonal polynomials). Moscow: Nauka Publ., 1976, 327 p. ISBN (3rd ed.): 978-5-9221-0406-7 .