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On incomplete
polynomials. II
Edward Barry Saff and Richard Steven Varga |
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Vol. 92 (1981), No. 1, 161–172
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Abstract |
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The approximation of xn by incomplete polynomials is studied, i.e., we consider the extremal problem
for the supremum norm on [0,1]. We show that, for k fixed, nkEn−k,k → 𝜀k as n →∞, where
A generalization of this result for the case of lacunary polynomial approximation is given, as well as inequalities for En−k,k and 𝜀k. Furthermore, we prove that for any polynomial P(t) of degree at most k, there holds for the supremum norm ∥e−tP(t)∥[0,+∞] = ∥e−tP(t)∥[0,2k]. |
![n ∑k n−j k
En −k,k = inf{∥x + djx ∥[0,1] : (d1,⋅⋅⋅ ,dk) ∈ R },n ≧ k,
j=1](a150x.png)
![k−1
𝜀 = inf{∥e−t(tk + ∑ a tj)∥ : (a,⋅⋅⋅ ,a ) ∈ Rk }.
k j=0 j [0,+∞ ] 0 k−1](a151x.png)
Mathematical Subject Classification 2000
Primary: 41A10
Secondary: 65K10
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Milestones
Received: 31 May 1979
Published: 1 January 1981
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Authors |
| Edward Barry Saff | |
| Richard Steven Varga | |
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