Geodesic
Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 2, pp. 157-168 
M. A. Komarov. Rate of interpolation of analytic functions with regularly decreasing coefficients by simple partial fractions. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 23 (2023) no. 2, pp. 157-168. http://geodesic.mathdoc.fr/item/ISU_2023_23_2_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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}|, $m=1,2,\dots$. 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, $0) 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.

[1] Danchenko V. I., Danchenko D. Ya., “Approximation by simplest fractions”, Mathematical Notes, 70:4 (2001), 502–507 | DOI | DOI | MR | Zbl

[2] Danchenko V. I., Komarov M. A., Chunaev P. V., “Extremal and approximative properties of simple partial fractions”, Russian Mathematics, 62:12 (2018), 6–41 | DOI | MR | Zbl

[3] Kosukhin O. N., On some non-traditional methods of approximation, related to complex polynomials, Diss. Cand. Sci. (Phiz. and Math.), M., 2005, 80 pp. (in Russian)

[4] Kondakova E. N., “Interpolation by simple partial fractions”, Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 9:2 (2009), 30–37 (in Russian) | DOI | MR

[5] Komarov M. A., “A criterion for the best uniform approximation by simple partial fractions in terms of alternance”, Izvestiya: Mathematics, 79:3 (2015), 431–448 | DOI | DOI | MR | Zbl

[6] Danchenko V. I., Chunaev P. V., “Approximation by simple partial fractions and their generalizations”, Journal of Mathematical Sciences, 176:6 (2011), 844–859 | DOI | MR | Zbl

[7] Chunaev P. V. On a nontraditional method of approximation, Proceedings of the Steklov Institute of Mathematics, 270:1 (2010), 278–284 | DOI | MR | Zbl

[8] Chunaev P. V., “Interpolation by generalized exponential sums with equal weights”, Journal of Approximation Theory, 254 (2020), 105397 | DOI | MR | Zbl

[9] Borodin P. A., “Approximation by simple partial fractions with constraints on the poles. II”, Sbornik: Mathematics, 207:3 (2016), 331–341 | DOI | DOI | MR | Zbl

[10] Komarov M. A., “On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary”, Izvestiya: Mathematics, 84:3 (2020), 437–448 | DOI | DOI | MR | MR | Zbl

[11] Duren P. L., Theory of $H^p$ spaces, Academic Press, New York–London, 1970, 258 pp. | MR

[12] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series, v. 1, Elementary Functions, Nauka, M., 1981, 800 pp. (in Russian) | MR

[13] Bateman H., Erdélyi A., Higher Transcendental Functions, v. 3, Elliptic and Automorphic Functions, Lamé and Mathieu functions, Nauka, M., 1967, 300 pp. (in Russian) | MR