Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
normal families; best $L_p$-approximation
normal families; best $L_p$-approximation
Summary:
For sequences of rational functions, analytic in some domain, a theorem of Montel’s type is proved. As an application, sequences of rational functions of the best $L_p$-approximation with an unbounded number of finite poles are considered.
References:
[1] S. N. Bernstein: On the distribution of zeros of polynomials to a continuous function positive on a real interval. Complex Works, Vol. I, Acad. Nauk UdSSR, 1952, pp. 443–451. (Russian)
[2] G. Baker, Jr., and P. Graves-Morris: Padé Approximants. Encyclopedia of Mathematics and Its Applications, Part I, Volume 13, 14. Addison-Wesley Publishing Company, Massachusetts, 1981.
[3] H.-P. Blatt, E. B. Saff and M. Simkani: Jentzsch-Szegő type theorems for the zeros of best approximants. J. London Math. Soc. 38 (1988), 307–316. MR 0966302
[4] H.-P. Blatt, P. Isenles and E. B. Saff: Remarks on behavior of zeros of polynomials of best approximating polynomials and rational functions. Algorithmus for Approximation, J. C. Mason, U. C. Cox (eds.), Inst. Math. Conf. Ser., New Ser. 10, Oxford Press, Oxford, 1987, pp. 437–445.
[5] D. Braess: On a conjecture of Meinardus on rational approximation of $e^x$, II. J. Approx. Theory 40 (1984), 375–379. DOI 10.1016/0021-9045(84)90012-1 | MR 0740650
[6] J. Gilewicz and W. Pleśniak: Distribution of zeros of sequences of polynomials. Ann. Polon. Math 30 (1993), 165–177. MR 1233780
[7] G. M. Golusin: Geometric Theory of Functions of a Complex Variable. Nauka, Moscow, 1966. (Russian)
[8] A. A. Gonchar: On uniform convergence of diagonal Padé approximant.
[9] A. A. Gonchar: A local conjecture for the uniqueness of analytic functions. Matem. Sbornik 89 (1972), 148–164. (Russian)
[10] H. Gonska and R. K. Kovacheva: An extension of Montel’s theorems to some rational approximating sequences. Bull. Soc. Lettres Lodz 21 (1996), 73–86. MR 1475304
[11] R. Grothmann and E. B. Saff: On the behavior of zeros and poles of best uniform polynomial and rational approximation. Nonlinear Numerical Methods and Rational Approximations, D. Reidel Publ. Co., Dordrecht, 1988, pp. 57–77. MR 1005351
[12] R. K. Kovacheva: On the behaviour of Chebyshev rational approximants with a fixed number of poles. Math. Balkanica 3 (1989), 244–256. MR 1048047
[13] R. K. Kovacheva: Diagonal rational Chebyshev approximants and holomorphic continuation of functions. Analysis 10 (1990), 147–161. DOI 10.1524/anly.1990.10.23.147 | MR 1074829 | Zbl 0734.41019
[14] R. K. Kovacheva: An analogue of Montel’s theorem to rational approximating sequences. Comp. Ren. Acad. Bulg. Scien. 50 (1997), 9–12. MR 1630480 | Zbl 0927.30026
[15] A. Kroo and J. Swetits: On density of interpolation points, a Kadec type theorem and Saff’s principle of contamination in $L_p$-approximation. Constr. Approx. 8 (1992), 87–103. DOI 10.1007/BF01208908 | MR 1142696
[16] I. P. Natanson: The Theory of Analytic Functions. Nauka, Moscow, 1950. (Russian)
[17] I. I. Privalov: Boundary Oroperties of Analytic Functions. Nauka, Moscow, 1950. (Russian)
[18] E. B. Saff and H. Stahl: Ray sequences of best rational approximants for $|x|^{\alpha }$. Canad. J. Math. 49 (1997), 1034–1065. DOI 10.4153/CJM-1997-052-3 | MR 1604134
[19] H. Stahl: Best uniform rational approximation of $|x|$ on $[-1, 1]$. Mat. Sb. 8 (1983), 85–118. MR 1187250
[20] A. F. Timan: Approximation of Real-Vaued Functions. GIFMAT, 1960. (Russian)
[21] J. L. Walsh: Interpolation and approximation by rational functions in the complex plane. AMS Colloquium Publications, Volume XX, 1960.
[22] J. L. Walsh: Overconvergence, degree of convergence and zeros of sequences of analytic functions. Duke Math. J. 13 (1946), 195–235. MR 0017797 | Zbl 0063.08149
Search
Partner of
