On a theorem of S. Bernstei

Ankeny, N. C.; Rivlin, T. J.

doi:10.2140/pjm.1955.5.849

Cited by 110 publications

(66 citation statements)

References 1 publication

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“…In [6] the first author unified the results of MacLane [5] and Chui [2 ] by proving In the present paper we show that if f(z) is bounded in D, then the C-polynomials Pn(z) of Theorem A can be chosen so that for all « we have (1) I Pn(z) I < 2M, z in D. Equality in (5) is attained for the polynomial P(z) = (z + l)3/8. It seems possible, therefore, that estimates sharper than (4) can be obtained for certain inside coefficients of P(z).…”

Section: Bounded Approximation By Polynomials Whose Zeros Lie On a CIsupporting

confidence: 56%

Bounded Approximation by Polynomials whose Zeros Lie on a Circle

Rubinstein¹,

Saff²

1971

Proceedings of the American Mathematical Society

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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.

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Section: Bounded Approximation By Polynomials Whose Zeros Lie On a CIsupporting

confidence: 56%

Bounded Approximation by Polynomials whose Zeros Lie on a Circle

Rubinstein¹,

Saff²

1971

Proceedings of the American Mathematical Society

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“…Theorem B [1], Under the conditions of Theorem A (1.6) \p(z)\i(R"+l)/2 for |z| = F>l. For k = 0 these should reduce to (1.3) and (1.4) respectively. While seeking the desired extension of (1.4) we have only been able to prove the following Theorem 1.…”

mentioning

confidence: 98%

Functions of Exponential Type Not Vanishing in a Half-Plane and Related Polynomials

Govil¹,

Rahman²

1969

Transactions of the American Mathematical Society

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“…Ankeny and Rivlin [1] considered polynomials not vanishing in the interior of the unit circle and obtained refinement of inequality (1.1) by proving that if p(z) = 0 in |z| < 1, then…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%