Abstract:
In this paper Vitushkin's localization scheme for uniform rational approximation of functions is further improved. Using this result new criteria for the approximability of functions by rational fractions in the uniform and Holder metrics on compact subsets of C are obtained.
\Bibitem{Par95}
\by P.~V.~Paramonov
\paper Some new criteria for uniform approximability of functions by rational fractions
\jour Sb. Math.
\yr 1995
\vol 186
\issue 9
\pages 1325--1340
\mathnet{http://mi.mathnet.ru/eng/sm70}
\crossref{https://doi.org/10.1070/SM1995v186n09ABEH000070}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1360189}
\zmath{https://zbmath.org/?q=an:0947.41011}
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https://doi.org/10.1070/SM1995v186n09ABEH000070
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This publication is cited in the following 20 articles:
M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Criteria for $C^m$-approximability of functions by solutions of homogeneous second-order elliptic equations on compact subsets of $\mathbb{R}^N$ and related capacities”, Russian Math. Surveys, 79:5 (2024), 847–917
Liming Yang, “Invertibility in weak-star closed algebras of analytic functions”, Journal of Functional Analysis, 285:11 (2023), 110143
Liming Yang, “Cauchy transform and uniform approximation by polynomial modules”, Journal of Mathematical Analysis and Applications, 523:1 (2023), 127004
M. Ya. Mazalov, “Uniform approximation of functions
by solutions of second order homogeneous strongly elliptic equations on compact sets in ${\mathbb{R}}^2$”, Izv. Math., 85:3 (2021), 421–456
P. V. Paramonov, “Criteria for $C^1$-approximability of functions on compact sets in ${\mathbb{R}}^N$, $N\geqslant 3$, by solutions of second-order homogeneous elliptic equations”, Izv. Math., 85:3 (2021), 483–505
M. Ya. Mazalov, “Approximation by polyanalytic functions in Hölder spaces”, St. Petersburg Math. J., 33:5 (2022), 829–848
M. Ya. Mazalov, “A criterion for uniform approximability of individual functions by solutions of second-order homogeneous elliptic equations with constant complex coefficients”, Sb. Math., 211:9 (2020), 1267–1309
Paramonov P.V. Tolsa X., “On C-1-Approximability of Functions By Solutions of Second Order Elliptic Equations on Plane Compact Sets and C-Analytic Capacity”, Anal. Math. Phys., 9:3 (2019), 1133–1161
P. V. Paramonov, “Criteria for the individual $C^m$-approximability of functions on compact subsets of $\mathbb R^N$ by solutions of second-order homogeneous elliptic equations”, Sb. Math., 209:6 (2018), 857–870
P.M. Gauthier, P.V. Paramonov, F. Sharifi, “Meromorphic tangential approximation on the boundary of closed sets in Riemann surfaces”, Journal of Approximation Theory, 232 (2018), 1
P. V. Paramonov, “New Criteria for Uniform Approximability by Harmonic Functions on Compact Sets in $\mathbb R^2$”, Proc. Steklov Inst. Math., 298 (2017), 201–211
M. Ya. Mazalov, P. V. Paramonov, “Criteria for $C^m$-approximability by bianalytic functions on planar compact sets”, Sb. Math., 206:2 (2015), 242–281
M. Ya. Mazalov, “A criterion for approximability by harmonic functions in Lipschitz spaces”, J. Math. Sci. (N. Y.), 194:6 (2013), 678–692
M. Ya. Mazalov, P. V. Paramonov, K. Yu. Fedorovskiy, “Conditions for $C^m$-approximability of functions by solutions of elliptic equations”, Russian Math. Surveys, 67:6 (2012), 1023–1068
M. Ya. Mazalov, “Criterion of uniform approximability by harmonic functions on compact sets in $\mathbb R^3$”, Proc. Steklov Inst. Math., 279 (2012), 110–154
M. Ya. Mazalov, “Uniform approximation problem for harmonic functions”, St. Petersburg Math. J., 23:4 (2012), 731–759
M. Ya. Mazalov, “A criterion for uniform approximability on arbitrary compact sets for solutions of elliptic equations”, Sb. Math., 199:1 (2008), 13–44
M. Ya. Mazalov, “Uniform approximations by bianalytic functions on arbitrary compact subsets of $\mathbb C$”, Sb. Math., 195:5 (2004), 687–709
J. Verdera, M. S. Mel'nikov, P. V. Paramonov, “$C^1$-approximation and extension of subharmonic functions”, Sb. Math., 192:4 (2001), 515–535
M. Ya. Mazalov, “Uniform Approximation of Functions Continuous on a Compact Subset of $\mathbb C$ and Analytic in Its Interior by Functions Bianalytic in Its Neighborhoods”, Math. Notes, 69:2 (2001), 216–231
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