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Scientific Part
Mathematics
Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions
M. A. Komarov Vladimir State University, 87 Gorky St., Vladimir 600000, Russia
Abstract:
We consider the problems of multiple interpolation of analytic functions $f(z)=f_0+f_1z+\dots$ in the unit disk with node $z=0$ by means of simple partial fractions (logarithmic derivatives of algebraic polynomials) with free poles and with all poles on the circle $|z|=1$. We obtain estimates of the interpolation errors under a condition of the form $|f_{m-1}|<C/\sqrt{m}$, $m=1,2,\dots$. More precisely, we assume that the moduli of the Maclaurin coefficients $f_m$ of a function $f$ do not exceed the corresponding coefficients $\alpha_m$ in the expansion of $a/\sqrt{1-x}$ ($-1<x<1$, $0<a\le a^*\approx 0.34$) in powers of $x$. To prove the estimates, the constructions of Padé simple partial fractions with free poles developed by V. I. and D. Ya. Danchenko (2001), O. N. Kosukhin (2005), V. I. Danchenko and P. V. Chunaev (2011) and the construction of interpolating simple partial fractions with poles on the circle developed by the author (2020) are used. Our theorems complement and improve a number of results of the listed works. Using properties of the sequence $\{\alpha_m\}$ it is possible to prove, in particular, that under the condition $|f_m|\le \alpha_m$ all the poles of the Padé simple partial fraction of a function $f$ lie in the exterior of the unit circle.
Key words:
simple partial fraction, rational approximation, multiple interpolation, analytic function, power sum.
Received: 23.03.2022 Accepted: 16.11.2022
Citation:
M. A. Komarov, “Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions”, Izv. Saratov Univ. Math. Mech. Inform., 23:2 (2023), 157–168
Linking options:
https://www.mathnet.ru/eng/isu975 https://www.mathnet.ru/eng/isu/v23/i2/p157
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Abstract page: | 97 | Full-text PDF : | 29 | References: | 25 |
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