O. E. Galkin, S. Yu. Galkina, I. Yu. Yastrebova

National Research University «Higher School of Economics» (Nizhny Novgorod, Russian Federation)

Abstract. Polynomials that are least-deviating from zero play an important role in the theory and practice of using numerical methods. They can be used to solve problems of optimizing the properties of various computational algorithms. So, our work is devoted to the study of polynomials that are least-deviating from zero on the ray $[0,+\infty)$ in the exponential norm. Such a norm for any $\alpha>0$ and any infinite subset $K\subset\mathbb{R}$ is defined by the equality $\|P\|_{\alpha,K}=\sup \limits_{x\in K} e^{-|x|^\alpha}|P(x)|$. In this article, we discuss the existence, uniqueness, and characterization of polynomials that deviate least from zero in the norm $\|\cdot\|_{\alpha,[0;+\infty)}$, derive a system of equations that such polynomials must obey, and reformulate the results of Mhaskar and Saff (1984), obtained for the norm $\|\cdot\|_{\alpha,\mathbb{R}}$, according to our kind of norm. Next, we approximately calculate the polynomials of the first and second degrees that deviate least from zero according to the norm $\|\cdot\|_{1,[0;+\ infty)}$. Our method is an alternative to Remez algorithm. In the calculations, we use the principle of contracting mappings, Newton's method and Halley's method. Our results are illustrated by pictures.

Key Words: exponential norm, best approximation polynomial, Bernstein-Chebyshev theorem

For citation: O. E. Galkin, S. Yu. Galkina, I. Yu. Yastrebova. On some properties of polynomials that are least-deviating from zero on the positive semi-axis in the exponential norm. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 28:1(2026), 11–30. DOI: https://doi.org/10.15507/2079-6900.28.202601.11-30

Information about the authors:

Oleg E. Galkin, Ph. D. (Phys. and Math.), Associate Professor, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: https://orcid.org/0000-0003-2085-572X, olegegalkin@ya.ru

Svetlana Yu. Galkina, Ph. D. (Phys. and Math.), Associate Professor, Department of Fundamental Mathematics, National Research University «Higher School of Economics» (25/12 B. Pecherskaya St., Nizhny Novgorod 603155, Russia), ORCID: http://orcid.org/0000-0002-2476-2275, svetlana.u.galkina@mail.ru

Irina Yu. Yastrebova, Senior Lecturer, Department of Applied Mathematics, National Research Lobachevsky State University of Nizhny Novgorod (23 Gagarin Av., Nizhny Novgorod 603022, Russia), ORCID: https://orcid.org/0009-0009-5991-7466, yastrebova@unn.ru