Extremal polynomials connected with Zolotarev polynomials

Authors

DOI:

https://doi.org/10.21638/11701/spbu01.2020.101

Abstract

Let two points a and b be given on the real axis, located to the right and left of the segment [−1, 1] respectively. The extremal problem is posed: find an algebraic polynomial of n-th degree, which at the point a takes value A, on the segment [−1, 1] does not exceed M in modulus and takes the largest possible value at b. This problem is related to the second problem of Zolotarev. In the article the set of values of the parameter A for which this problem has a unique solution is indicated, and an alternance characteristic of this solution is given. The behavior of the solution with respect to the parameter A is studied. It turns out that for some A the solution can be obtained with the help of the Chebyshev polynomial, while for all other admissible A — with the help of the Zolotarev polynomial.

Keywords:

extremal properties of polynomials, alternance, Chebyshev polynomials, Zolotarev polynomials

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References

Литература

Золотарёв Е. И. Приложение эллиптических функций к вопросам о функциях, наименее и наиболее отклоняющихся от нуля // В кн.: Золотарёв Е. И. Полное собрание сочинений. Выпуск второй. Л.: Изд-во АН СССР, 1932. С. 1–59.

Агафонова И. В., Малозёмов В. Н. Экстремальные полиномы, связанные с полиномами Золотарёва // Докл. Академии наук. 2016. Т. 5. Вып. 467. С. 255–256.

Мысовских И. П. Лекции по методам вычислений. 2-е изд., перераб. и доп. СПб.: Изд-во С.-Петерб. ун-та, 1998.

Даугавет В. А., Малозёмов В. Н. Нелинейные задачи аппроксимации // В кн.: Современное состояние теории исследования операций. М.: Наука, 1979. С. 336–363.

Малозёмов В. Н., Тамасян Г.Ш. Этюд на тему полиномиальной фильтровой задачи (n = 3) // В кн.: Избранные лекции по экстремальным задачам. Часть вторая. СПб.: Изд-во ВВМ, 2017. C. 305–315. URL: http://www.apmath.spbu.ru/cnsa/reps15.shtml#0312 (дата обращения: 26.05.2019).


Reference

Zolotarev E. I., Application of elliptic functions to questions of functions deviating least and most from zero, In: Collected works 2, 1–59 (Izdat. Akad. Nauk SSSR, Moscow, 1932). (In Russian)

Agafonova I. V., Malozemov V. N., “Extremal polynomials connected with Zolotarev polynomials”, Dokl. Akad. Nauk 467(5), 255–256 (2016). (In Russian)

Mysovskih I. P., Lectures on Numerical Methods (St. Petersburg Univ. Press, St. Petersburg, 1998). (In Russian)

Daugavet V. A., Malozemov V. N., Nonlinear approximation problems, in: The State-of-the-Art of Operations Research Theory, 336–363 (N. N. Moiseev (ed.), Nauka Publ., Moscow, 1979). (In Russian)

Malozemov V. N., Tamasyan G. Sh., An etude on the polynomial filter problem (n = 3), In: Selected Lectures on Extremal Problems. Part II, 305–315 (VVM Publ., St. Petersburg, 2017). Available at: http://www.apmath.spbu.ru/cnsa/reps15.shtml#0312 (accessed: May 26, 2019). (In Russian)

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Published

2020-05-12

How to Cite

Agafonova, I. V., & Malozemov, V. N. (2020). Extremal polynomials connected with Zolotarev polynomials. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(1), 3–14. https://doi.org/10.21638/11701/spbu01.2020.101

Issue

Vol. 7 No. 1 (2020)

Section

Mathematics

License

Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.