We study extremal properties of simple partial fractions ρn (i.e., the logarithmic derivatives of algebraic polynomials of degree n) on a segment and on a circle. We prove that for any a > 1 the poles of a fraction ρn whose sup norm does not exceed ln(1 + a−n) on [−1, 1] lie in the exterior of the ellipse with foci ±1 and sum of half-axes a. For a real-valued analytic function f bounded in the ellipse with \( a=3+2\sqrt{2} \) we show that if a real-valued simple partial fraction of order not greater than n is least deviating from f in the C([−1, 1])-metric, then such a fraction is unique and is characterized by an alternance of n + 1 points in the segment [−1, 1].
Similar content being viewed by others
References
V. I. Danchenko, M. A. Komarov, and P. V. Chunaev, “Extremal and approximative properties of simple partial fractions,” Russ. Math. 62, No. 12, 6–41 (2018).
V. I. Danchenko, “Estimates of derivatives of simplest fractions and other questions.” Sb. Math. 197, No. 4, 505–524 (2006).
V. I. Danchenko and A. E. Dodonov, “Estimates for Lp-norms of simple partial fractions,” Russ. Math. 58, No. 6, 6–15 (2014).
S. N. Bernstein, Extremal Properties of Polynomials and the Best Approximation of Continuous Functions of One Real Variable [in Russian], GONTI, Leningrad–Moscow (1937).
V. I. Danchenko and E. N. Kondakova, “Chebyshev’s alternance in the approximation of constants by simple partial fractions,” Proc. Steklov Inst. Math. 270, 80–90 (2010).
M. A. Komarov, “An example of non-uniqueness of a simple partial fraction of the best uniform approximation” Russ. Math. 57, No. 9, 22–30 (2013).
M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance,” Izv. Math. 79, No. 3, 431–448 (2015).
M. A. Komarov, “A criterion for the best uniform approximation by simple partial fractions in terms of alternance. II,” Izv. Math. 81, No. 3, 568–591 (2017).
O. N. Kosukhin, “Approximation properties of the most simple fractions,” Mosc. Univ. Math. Bull. 56, No. 4, 36–40 (2001).
M. A. Komarov, “Estimates of the best approximation of polynomials by simple partial fractions,” Math. Notes 104, No. 6, 848–858 (2018).
R. J. Duffin and A. C. Schaeffer, “Some properties of functions of exponential type,” Bull. Amer. Math. Soc. 44, No. 4, 236–240 (1938).
P. V. Chunaev, “Least deviation of logarithmic derivatives of algebraic polynomials from zero,” J. Approx. Theory 185, 98–106 (2014).
A. Aziz and Q. M. Dawood, “Inequalities for a polynomial and its derivative,” J. Approx. Theory 54, 306–313 (1988).
T. Sheil-Small, Complex Polynomials, Cambridge Univ. Press, Cambridge (2002).
K. K. Dewan and N. K. Govil, “An inequality for the derivative of self-inversive polynomials,” Bull. Aust. Math. Soc. 27, 403–406 (1983).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 3-9.
Rights and permissions
About this article
Cite this article
Komarov, M.A. Extremal Properties of Logarithmic Derivatives of Polynomials. J Math Sci 250, 1–9 (2020). https://doi.org/10.1007/s10958-020-04991-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-020-04991-y