Abstract:
It is shown that if a compact set KK not separating the plane C lies in the union ˆE∖E of the bounded components of the complement of another compact set E, then the simple partial fractions
(the logarithmic derivatives of polynomials) with poles in E are dense in the space AC(K) of functions that are continuous on K and analytic in its interior. It is also shown that if a compact set K with connected complement
lies in the complement C∖¯D of the closure of a doubly connected domain D⊂¯C with bounded connected components of the boundary E+ and E−, then the differences r1−r2 of the simple partial fractions such that r1 has its poles in E+ and r2 has its poles in E− are dense in the space AC(K).
Bibliography: 9 titles.
Keywords:
simple partial fractions, uniform approximation, restriction on the poles, neutral distribution, condenser.
This research was carried out with the financial support of the Russian Foundation for Basic Research (grant nos. 14-01-00510, 14-01-91158, and 15-01-08335) and the Dmitry Zimin Dynasty Foundation.
This publication is cited in the following 18 articles:
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M. A. Komarov, “Density of Simple Partial Fractions with Poles on a Circle in Weighted Spaces for a Disk and a Segment”, Vestnik St.Petersb. Univ.Math., 57:1 (2024), 62
K. Shklyaev, “Approximation by sums of shifts and dilations of a single function and neural networks”, Journal of Approximation Theory, 291 (2023), 105915
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M. A. Komarov, “O skorosti interpolyatsii naiprosteishimi drobyami analiticheskikh funktsii s regulyarno ubyvayuschimi koeffitsientami”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 23:2 (2023), 157–168
P. A. Borodin, K. S. Shklyaev, “Density of quantized approximations”, Russian Math. Surveys, 78:5 (2023), 797–851
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P. A. Borodin, K. S. Shklyaev, “Approximation by simple partial fractions in unbounded domains”, Sb. Math., 212:4 (2021), 449–474
P. A. Borodin, “Greedy approximation by arbitrary set”, Izv. Math., 84:2 (2020), 246–261
M. A. Komarov, “On the rate of approximation in the unit disc of $H^1$-functions by logarithmic derivatives of polynomials with zeros on the boundary”, Izv. Math., 84:3 (2020), 437–448
M. A. Komarov, “Approximation to constant functions by electrostatic fields due to electrons and positrons”, Lobachevskii J. Math., 40:1, SI (2019), 79–84
M. A. Komarov, “A lower bound for the $L_2[-1,\,1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle”, Probl. anal. Issues Anal., 8(26):2 (2019), 67–72
P. A. Borodin, S. V. Konyagin, “Convergence to zero of exponential sums with positive integer coefficients and approximation by sums of shifts of a single function on the line”, Anal. Math., 44:2 (2018), 163–183
M. A. Komarov, “On approximation by special differences of simplest fractions”, St. Petersburg Math. J., 30:4 (2019), 655–665
P. A. Borodin, “Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk”, Math. Notes, 104:1 (2018), 3–9
V. I. Danchenko, M. A. Komarov, P. V. Chunaev, “Extremal and approximative properties of simple partial fractions”, Russian Math. (Iz. VUZ), 62:12 (2018), 6–41
P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. Math., 81:6 (2017), 1080–1094
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