Best Polynomial Approximation with Linear Constraints
Keyword(s):
AbstractLet 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).
1977 ◽
Vol 16
(3)
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pp. 361-369
1982 ◽
Vol 19
(04)
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pp. 851-857
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2013 ◽
Vol 37
(8)
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pp. 1219-1231
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2019 ◽
Vol 18
(02)
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pp. 1950057
2004 ◽
Vol 2004
(58)
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pp. 3103-3116
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