Behavior of zeros of polynomials of near best approximation

Blatt, Hans-Peter; Saff, Edward B.

doi:10.1016/0021-9045(86)90069-9

Cited by 31 publications

(22 citation statements)

References 4 publications

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“…The equivalence of the assertions 1 and 3 of Theorem 3 was proved by BLATT and SAVE [2] under the assumptions that c~\ E is connected and each point of the boundary of Eoo is regular with respect to the Dirichlet problem. The proof of implication 3 =~ 1 is similar to that of Blatt and Saff but it is presented here for further references.…”

Section: Ilf -P Ll < N R Y~mentioning

confidence: 85%

On zeros of polynomials of best approximation to holomorphic andC ? functions

Wójcik

1988

Monatshefte f�r Mathematik

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Abstract. Let E be a compact subset of the complex plane cg such that Leja's extremal function L~ for E is continuous. If almost all zeros of the polynomials of best approximation to a function f~C(E) are outside the set E R = {z~Cg:LE(z)< R}, for some R> 1, then f is extendible to a holomorphic function in E R. If the zeros of n-th polynomial of best approximation to fare outside Eg. and the sequence {R, -n} rapidly decreases to zero thenfcan be extended to a C ~ function on ~2. In this paper Bernstein's theorem is proved in the case of uniform approximation on a compact subset E of cg such that Leja's extremal function Le associated with E is continuous (see Corollary 4). A similar result for the polynomials of best approximation to C ~ functions on sufficiently regular compact subsets of cg is also proved (Theorem 9).Let E be a compact subset of the complex plane. Denote by Eoo the unbounded connected component of cg \ E. Throughout the paper we assume that E is not thin at any point of the boundary of Eo~ (or: each

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Section: Ilf -P Ll < N R Y~mentioning

confidence: 85%

On zeros of polynomials of best approximation to holomorphic andC ? functions

Wójcik

1988

Monatshefte f�r Mathematik

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“…In their paper [2], Blatt and Saff prove the following results. Then G E (z) is an``exact harmonic majorant'' ( for definition see [2]) for [ p d (z)] on C&E. …”

Section: Introductionmentioning

confidence: 87%

Some Applications of the Robin Function to Multivariable Approximation Theory

Bloom

1998

Journal of Approximation Theory

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Let E/C n be compact, regular, and polynomially convex with pluricomplex Green function V E . Given a sequence of polynomials [ p j ] j=1, 2, ... , the first result is a condition for ( limThe condition involves the Robin function of E and the highest order homogeneous terms of the p j and generalizes one-variable results of Blatt Saff. A second result gives a necessary and sufficient condition for f, the uniform limit of polynomials on E, to extend holomorphically to E R =[z | V E (z)1. The condition involves highest order homogeneous terms of best approximating polynomials to f and the Robin function of E and extends results of Szczepan ski.1998 Academic Press

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“…This is a known fact established in [4]; it was shown there that R n (z) → z uniformly inside the right half-plane {z, Re z > 0} and to −z inside the left half-plane {z, Re z < 0}.…”

Section: This Implies (Ii)mentioning

confidence: 95%

“…It was proved in [4] that both the poles and the zeros of the rational function R n,n of best uniform approximation lie interlacing each other on the imaginary axis and go, as n → ∞, to infinity. Hence, condition (i) holds.…”

Section: Theorem 8 ([14]mentioning

confidence: 99%