Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)

For citation:

Lukashov А. L., Tyshkevich S. V. Extremal Rational Functions on Several Arcs of the Unit Circle with Zeros on these Arcs. Izv. Sarat. Univ. Math. Mech. Inform., 2009, vol. 9, iss. 1, pp. 8-13. DOI: 10.18500/1816-9791-2009-9-1-8-13

Published online: 
Full text:
(downloads: 46)

Extremal Rational Functions on Several Arcs of the Unit Circle with Zeros on these Arcs

Lukashov А. L., Saratov State University
Tyshkevich Sergey Viktorovich, Saratov State University

The solution of anextrema lproblem aboutrational function with fixed denominator and leading coefficient of nominator which is deviatedleast from zero on several arcs of the unit circle is given under restrictions on the location of zeros and additional conditions on mutual position of the arcs and zeros of denominator. The extremal function is represented in terms of the density of harmonic measure.

  1. Смирнов В.И., Лебедев Н.А. Конструктивная теория функций комплексного переменного. М.; Л.: Наука, 1964.
  2. Thiran J.-P., Detaille C. Polynomials on Circular Arcs in the Complex Plane // Progress in Approximation Theory. Boston; QA: Academic Press. 1991. P. 771–786.
  3. Widom H. Extremal polinomials associated with a system of curves in the complex plane // Adv. Math. 1969. V. 3. P. 127–232.
  4. Лукашов А.Л. Неравенства для производных рациональных функций // Изв. РАН. Сер. матем. 2004. Т. 68, № 3. С. 115–138.
  5. Тышкевич С.В. О чебышёвских полиномах на дугах окружности // Мат. заметки. 2007. Т. 81, вып. 6. С. 952–954.