Bounded approximation by polynomials whose zeros lie on a circle
Authors:
Zalman Rubinstein and E. B. Saff
Journal:
Proc. Amer. Math. Soc. 29 (1971), 482486
MSC:
Primary 30.70
MathSciNet review:
0277730
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In a recent paper the first author gave an explicit construction of a sequence of polynomials having their zeros on the unit circumference which converge boundedly to a given bounded zerofree analytic function in the unit disk. In this paper we find the best possible uniform bound for such approximating polynomials and construct a sequence for which this bound is attained. The method is also applied to approximation of an analytic function in the unit disk by rational functions whose poles lie on the unit circumference. Some open problems are discussed.
 [1]
N.
C. Ankeny and T.
J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math.
5 (1955), 849–852. MR 0076020
(17,833e)
 [2]
Charles
Kamtai Chui, Bounded approximation by polynomials
with restricted zeros, Bull. Amer. Math.
Soc. 73 (1967),
967–972. MR 0218584
(36 #1669), http://dx.doi.org/10.1090/S000299041967118676
 [3]
W.
K. Hayman, Research problems in function theory, The Athlone
Press University of London, London, 1967. MR 0217268
(36 #359)
 [4]
Peter
D. Lax, Proof of a conjecture of P. Erdös
on the derivative of a polynomial, Bull. Amer.
Math. Soc. 50
(1944), 509–513. MR 0010731
(6,61f), http://dx.doi.org/10.1090/S000299041944081779
 [5]
Gerald
R. Mac Lane, Polynomials with zeros on a rectifiable Jordan
curve, Duke Math. J. 16 (1949), 461–477. MR 0030594
(11,20e)
 [6]
Zalman
Rubinstein, On the approximation by
𝐶polynomials, Bull. Amer. Math.
Soc. 74 (1968),
1091–1093. MR 0232003
(38 #329), http://dx.doi.org/10.1090/S000299041968120579
 [7]
Maynard
Thompson, Approximation of bounded analytic functions on the
disc, Nieuw Arch. Wisk. (3) 15 (1967), 49–54.
MR
0217305 (36 #396)
 [1]
 N. C. Ankeny and T. J. Rivlin, On a theorem of S. Bernstein, Pacific J. Math. 5 (1955), 849852. MR 17, 833. MR 0076020 (17:833e)
 [2]
 Charles KamTai Chui, Bounded approximation by polynomials with restricted zeros. Bull. Amer. Math. Soc. 73 (1967), 967972. MR 36 #1669. MR 0218584 (36:1669)
 [3]
 W. K. Hayman, Research problems in function theory, Athlone Press, London, 1967. MR 36 #359. MR 0217268 (36:359)
 [4]
 P. D. Lax, Proof of a conjecture of P. Erdös on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509513. MR 6, 61. MR 0010731 (6:61f)
 [5]
 G. R. MacLane, Polynomials with zeros on a rectifiable Jordan curve, Duke Math. J. 16 (1949), 461477. MR 11, 20. MR 0030594 (11:20e)
 [6]
 Z. Rubinstein, On the approximation by Cpolynomials, Bull. Amer. Math. Soc. 74 (1968), 10911093. MR 38 #329. MR 0232003 (38:329)
 [7]
 M. Thompson, Approximation of bounded analytic functions on the disc, Nieuw Arch. Wisk. (3) 15 (1967), 4954. MR 36 #396. MR 0217305 (36:396)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
30.70
Retrieve articles in all journals
with MSC:
30.70
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919710277730X
PII:
S 00029939(1971)0277730X
Keywords:
Bounded approximation by polynomials,
Cpolynomials,
coefficients of polynomials,
rational functions,
logarithmic derivative
Article copyright:
© Copyright 1971
American Mathematical Society
