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

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6850(online) ISSN 0002-9947(print)

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

Authors: S. Kouchekian and V. A. Prokhorov
Journal: Trans. Amer. Math. Soc. 361 (2009), 2649-2663
MSC (2000): Primary 41A20, 30E10; Secondary 47B35
Posted: December 4, 2008
MathSciNet review: 2471933
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be an arbitrary compact subset of the extended complex plane $\overline {\mathbb{C}}$ with nonempty interior. For a function continuous on and analytic in the interior of denote by $\rho_n(f; E)$ the least uniform deviation of on from the class of all rational functions of order at most . In this paper we show that if is not a rational function and if is an arbitrary compact subset of the interior of 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.

1. 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 (45 #7505)
2. Thomas Bagby, On interpolation by rational functions, Duke Math. J. 36 (1969), 95–104. MR 0241655 (39 #2994)
3. Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970. MR 0268655 (42 #3552)
4. 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 (99f:15001)
5. I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR 0246142 (39 #7447)
6. 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 (40 #308)
7. 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.
8. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027 (50 #2520)
9. 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 (88e:46031)
10. I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765 (13,926h)
11. 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 (94b:47035), http://dx.doi.org/10.1070/SM1994v078n01ABEH003459
12. 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 (94h:41029), http://dx.doi.org/10.1070/SM1994v078n01ABEH003736
13. 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 (2005i:41021)
14. V. A. Prokhorov, On best rational approximation of analytic functions, J. Approx. Theory 133 (2005), no. 2, 284–296. MR 2129484 (2006d:41016), http://dx.doi.org/10.1016/j.jat.2004.12.007
15. V. A. Prokhorov and M. Putinar, Compact Hankel forms on planar domains (manuscript).
16. 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 (99h:31001)
17. Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153 (2006f:47086)
18. 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 (20 #114)
19. J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Fourth edition. American Mathematical Society Colloquium Publications, Vol. XX, American Mathematical Society, Providence, R.I., 1965. MR 0218588 (36 #1672b)
20. 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 0419783 (54 #7801)

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|