On estimates for the ratio of errors in best rational approximation of analytic functions

Kouchekian, S.; Prokhorov, V.

doi:10.1090/S0002-9947-08-04628-X

On estimates for the ratio of errors in best rational approximation of analytic functions
HTML articles powered by AMS MathViewer

by S. Kouchekian and V. A. Prokhorov PDF

Trans. Amer. Math. Soc. 361 (2009), 2649-2663 Request permission

Abstract:

Let $E$ be an arbitrary compact subset of the extended complex plane $\overline {\mathbb C}$ with nonempty interior. For a function $f$ continuous on $E$ and analytic in the interior of $E$ denote by $\rho _n(f; E)$ the least uniform deviation of $f$ on $E$ from the class of all rational functions of order at most $n$. In this paper we show that if $f$ is not a rational function and if $K$ is an arbitrary compact subset of the interior of $E,$ then $\prod _{k=0}^n (\rho _k(f; K) /\rho _k(f; E) ),$ the ratio of the errors in best rational approximation, converges to zero geometrically as $n \to \infty$ and the rate of convergence is determined by the capacity of the condenser $(\partial E, K)$. In addition, we obtain results regarding meromorphic approximation and sharp estimates of the Hadamard type determinants.

References

V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn, Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem, Mat. Sb. (N.S.) 86(128) (1971), 34–75 (Russian). MR 0298453
Thomas Bagby, On interpolation by rational functions, Duke Math. J. 36 (1969), 95–104. MR 241655
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
F. R. Gantmacher, The theory of matrices. Vol. 1, AMS Chelsea Publishing, Providence, RI, 1998. Translated from the Russian by K. A. Hirsch; Reprint of the 1959 translation. MR 1657129
I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039, DOI 10.1090/mmono/026
A. A. Gonchar, Rational approximation of analytic functions, Linear and Complex Analysis Problem Book (V. P. Havin [Khavin] et al., editors) Lecture Notes in Math., vol. 1043, Springer–Verlag, Berlin, 1984, 471–474.
N. S. Landkof, Foundations of modern potential theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy. MR 0350027, DOI 10.1007/978-3-642-65183-0
O. G. Parfënov, Estimates for singular numbers of the Carleson embedding operator, Mat. Sb. (N.S.) 131(173) (1986), no. 4, 501–518 (Russian); English transl., Math. USSR-Sb. 59 (1988), no. 2, 497–514. MR 881910, DOI 10.1070/SM1988v059n02ABEH003148
I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
V. A. Prokhorov, A theorem of Adamyan-Arov-Kreĭn, Mat. Sb. 184 (1993), no. 1, 89–104 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 77–90. MR 1211367, DOI 10.1070/SM1994v078n01ABEH003459
V. A. Prokhorov, Rational approximation of analytic functions, Mat. Sb. 184 (1993), no. 2, 3–32 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 78 (1994), no. 1, 139–164. MR 1214941, DOI 10.1070/SM1994v078n01ABEH003736
V. A. Prokhorov, On estimates of Hadamard type determinants and rational approximation, Advances in constructive approximation: Vanderbilt 2003, Mod. Methods Math., Nashboro Press, Brentwood, TN, 2004, pp. 421–432. MR 2089942
V. A. Prokhorov, On best rational approximation of analytic functions, J. Approx. Theory 133 (2005), no. 2, 284–296. MR 2129484, DOI 10.1016/j.jat.2004.12.007
V. A. Prokhorov and M. Putinar, Compact Hankel forms on planar domains (manuscript).
Edward B. Saff and Vilmos Totik, Logarithmic potentials with external fields, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 316, Springer-Verlag, Berlin, 1997. Appendix B by Thomas Bloom. MR 1485778, DOI 10.1007/978-3-662-03329-6
Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
G. C. Turmarkin and S. Ja. Havinson, On the definition of analytic functions of class $E_{p}$ in multiply connected domains, Uspehi Mat. Nauk (N.S.) 13 (1958), no. 1(79), 201-206 (Russian). MR 0093590
J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1960. MR 0218587
V. P. Zaharjuta and N. I. Skiba, Estimates of the $n$-widths of certain classes of functions that are analytic on Riemann surfaces, Mat. Zametki 19 (1976), no. 6, 899–911 (Russian). MR 419783