Transactions of the American Mathematical Society

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

Author(s): S. Kouchekian; V. A. Prokhorov
Journal: Trans. Amer. Math. Soc. 361 (2009), 2649-2663.
MSC (2000): Primary 41A20, 30E10; Secondary 47B35
Posted: December 4, 2008
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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

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. Adamyan, D. Z. Arov, and M. G. Kreın, Analytic properties of Schmidt pairs, Hankel operators, and the generalized Schur-Takagi problem, Mat. Sb. 86 (128) (1971), 34-75; English transl. in Math. USSR Sb. 15 (1971). MR 0298453 (45:7505)
2.: T. Bagby, On interpolation by rational functions, Duke Math. J. 36 (1969), 95-104. MR 0241655 (39:2994)
3.: P. L. Duren, Theory of spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
4.: F. R. Gantmacher, The theory of matrices, 4th ed., ``Nauka'', Moscow, 1988; English transl., AMS Chelsea Publishing, Providence, RI, 1998. MR 1657129 (99f:15001)
5.: I. Ts. Gokhberg [Israel Gohberg] and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators in Hilbert space, ``Nauka'', Moscow, 1965; English transl., Amer. Math. Soc., Providence, RI, 1969. MR 0246142 (39:7447)
6.: G. M. Goluzin, Geometric theory of functions of a complex variable, 2nd ed., ``Nauka'', Moscow, 1966; English transl., Amer. Math. Soc., Providence, RI, 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, ``Nauka'', Moscow, 1966; English transl., Springer-Verlag, Berlin, 1972. MR 0350027 (50:2520)
9.: O. G. Parfenov, Estimates of the singular numbers of a Carleson operator, Mat. Sb. 131 (173) (1986), 501-518; English transl. in Math. USSR Sb. 59 (1988). MR 881910 (88e:46031)
10.: I. I. Privalov, Boundary properties of analytic functions, 2nd ed., GITTL, Moscow, 1950; German transl., VEB Deutscher Verlag Wiss., Berlin, 1956. MR 0047765 (13:926h)
11.: V. A. Prokhorov, On a theorem of Adamyan, Arov, and Kreĭn, Mat. Sb. 184 (1993), 89-104; English transl. in Russian Acad. Sci. Sb. Math. 78 (1994). MR 1211367 (94b:47035)
12.: V. A. Prokhorov, Rational approximation of analytic function, Mat. Sb. 184 (1993), 3-32; English transl. in Russian Acad. Sci. Sb. Math. 78 (1994). MR 1214941 (94h:41029)
13.: V. A. Prokhorov, On estimates of Hadamard type determinants and rational approximation, Advances in Constructive Approximation (Nashville, TN, 2003), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, 2004. MR 2089942 (2005i:41021)
14.: V. A. Prokhorov, On best rational approximation of analytic functions, J. Approx. Theory 133 (2005), 284-296. MR 2129484 (2006d:41016)
15.: V. A. Prokhorov and M. Putinar, Compact Hankel forms on planar domains (manuscript).
16.: E. B. Saff and V. Totik, Logarithmic potentials with external fields, Springer-Verlag, Heidelberg, 1997. MR 1485778 (99h:31001)
17.: B. Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs vol. 120, Amer. Math. Soc., Providence, RI, 2005. MR 2154153 (2006f:47086)
18.: G. Ts. Tumarkin and S. Ya. Khavinson, On the definition of analytic functions of class in multiply connected domains, Uspekhi Mat. Nauk 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, 2nd ed., Amer. Math. Soc., Providence, RI, 1956. MR 0218588 (36:1672b)
20.: V. P. Zaharjuta and N. T. Skiba, Estimates of the -widths of certain classes of functions that are analytic on Riemann surfaces, Mat. Zametki 19 (1976), no. 6, 899-911. MR 0419783 (54:7801)

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S. Kouchekian
Affiliation: Department of Mathematics & Statistics, University of South Florida, Tampa, Florida 33620-5700
Email: skouchek@cas.usf.edu

V. A. Prokhorov
Affiliation: Department of Mathematics & Statistics, ILB 325, University of South Alabama, Mobile, Alabama 36668
Email: prokhoro@jaguar1.usouthal.edu

DOI: 10.1090/S0002-9947-08-04628-X
PII: S 0002-9947(08)04628-X
Keywords: Rational approximation, singular number, meromorphic approximation, Hadamard type determinants
Received by editor(s): October 2, 2005
Received by editor(s) in revised form: August 3, 2007
Posted: December 4, 2008
Additional Notes: The first author was supported in part by the National Science Foundation grant DMS-0500916
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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