Bounded approximation by polynomials whose zeros lie on a circle

Authors:
Zalman Rubinstein and E. B. Saff

Journal:
Proc. Amer. Math. Soc. **29** (1971), 482-486

MSC:
Primary 30.70

DOI:
https://doi.org/10.1090/S0002-9939-1971-0277730-X

MathSciNet review:
0277730

Full-text 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 zero-free 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**17**, 833. MR**0076020 (17:833e)****[2]**Charles Kam-Tai Chui,*Bounded approximation by polynomials with restricted zeros*. Bull. Amer. Math. Soc.**73**(1967), 967-972. 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), 509-513. MR**6**, 61. MR**0010731 (6:61f)****[5]**G. R. MacLane,*Polynomials with zeros on a rectifiable Jordan curve*, Duke Math. J.**16**(1949), 461-477. MR**11**, 20. MR**0030594 (11:20e)****[6]**Z. Rubinstein,*On the approximation by C-polynomials*, Bull. Amer. Math. Soc.**74**(1968), 1091-1093. MR**38**#329. MR**0232003 (38:329)****[7]**M. Thompson,*Approximation of bounded analytic functions on the disc*, Nieuw Arch. Wisk. (3)**15**(1967), 49-54. MR**36**#396. MR**0217305 (36:396)**

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:
https://doi.org/10.1090/S0002-9939-1971-0277730-X

Keywords:
Bounded approximation by polynomials,
*C*-polynomials,
coefficients of polynomials,
rational functions,
logarithmic derivative

Article copyright:
© Copyright 1971
American Mathematical Society