Extremal Properties of Logarithmic Derivatives of Polynomials
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
EXTREMAL PROPERTIES OF LOGARITHMIC DERIVATIVES OF POLYNOMIALS M. A. Komarov A. G. and N. G. Stoletov Vladimir State University 87, Gor’kogo St., Vladimir 600000, Russia [email protected]
UDC 517.538
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 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]. Bibliography: 15 titles.
In approximation theory, by a simple partial rational fraction of order n one means the logarithmic derivative ρn = Q /Q of an algebraic polynomial of degree n, i.e., ρn (z) =
n k=1
1 z − zk
(n = 1, 2, . . . ),
ρ0 (z) ≡ 0,
where the poles zk are not necessarily distinct from the geometric point of view. Many extremal problems are connected with simple partial fractions. Methods of approximation and interpolation by simple partial fractions and their generalizations are systematically developed (cf. the survey [1]). In this paper, we continue the study of some important problems connected with extremal and approximation properties of simple partial rational fractions.
1
Criterion for Simple Partial Fractions with Poles in the Exterior of an Ellipse
In the theory of approximation of functions by simple partial fractions (cf. Section 2), the remoteness of the poles of fractions from a given set K, where the fractions are normalized, plays an important role. In this paper, we consider the case K = [−1, 1]. Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 3-9. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0001
1
For each a > 1 in the plane of complex variable z, we consider the ellipse Ca with foci z = 1 and z = −1 and sum of half-axes a 1 1 i 1 z= a+ cos t + a− sin t, 0 t < 2π (a > 1). (1.1) 2 a 2 a It is known (cf. [2, Section 8.1]) that all the poles of a simple partial rational fraction ρn of order n lie in the exterior of the ellipse (1.1) if ρn (2a)−n−1 (here and below, · denotes the sup norm in [−1, 1]). We establish a stronger result by a rather simple method. Theorem 1.1. The poles of a simple partial fraction ρn of order n lie in the exterior of the ellipse (1.1) if (1.2) ρn ln(1 + a−n ) (a > 1). Remark 1.1. For large n the majorant is approximately a−n . The condition (1.2) is sharp in the sense that for any a > 1 there exists a sequence of simple partial fractions ρ∗n of order n = 1, 2, . . . whose poles unboundedly approach the ellipse Ca as n → ∞, whereas ρ∗n = a−n . Indeed, we use the following
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