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Best Polynomial Approximation with Linear Constraints

Published online by Cambridge University Press:  20 November 2018

K. Pan  and
E. B. Saff
K. Pan
Affiliation:
Department of Mathematics, University of California, Riverside, California 92521, USA
E. B. Saff
Affiliation:
Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Tampa, Florida 33620, USA
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Abstract

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Let A be a (k + 1) × (k + 1) nonzero matrix. For polynomials p ∈ Pn, set and . Let E ⊂ C be a compact set that does not separate the plane and f be a function continuous on E and analytic in the interior of E. Set and . Our goal is to study approximation to f on E by polynomials from Bn(A). We obtain necessary and sufficient conditions on the matrix A for the convergence En(A,f) → 0 to take place. These results depend on whether zero lies inside, on the boundary or outside E and yield generalizations of theorems of Clunie, Hasson and Saff for approximation by polynomials that omit a power of z. Let be such that . We also study the asymptotic behavior of the zeros of and the asymptotic relation between En(f) and En(A,f).

Keywords

41A29
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 6 , 01 December 1992 , pp. 1289 - 1302
Copyright
Copyright © Canadian Mathematical Society 1992

References

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