Vol. 92, No. 1, 1981

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ISSN: 0030-8730
On incomplete polynomials. II

Edward Barry Saff and Richard Steven Varga

Vol. 92 (1981), No. 1, 161–172
Abstract

The approximation of xn by incomplete polynomials is studied, i.e., we consider the extremal problem

              n  ∑k    n−j                   k
En −k,k = inf{∥x +    djx   ∥[0,1] : (d1,⋅⋅⋅ ,dk) ∈ R },n ≧ k,
j=1

for the supremum norm on [0,1]. We show that, for k fixed, nkEnk,k 𝜀k as n →∞, where

                k−1
𝜀  = inf{∥e−t(tk + ∑ a tj)∥      : (a,⋅⋅⋅ ,a ) ∈ Rk }.
k              j=0 j   [0,+∞ ]   0     k−1

A generalization of this result for the case of lacunary polynomial approximation is given, as well as inequalities for Enk,k and 𝜀k. Furthermore, we prove that for any polynomial P(t) of degree at most k, there holds for the supremum norm etP(t)[0,+] = etP(t)[0,2k].

Mathematical Subject Classification 2000
Primary: 41A10
Secondary: 65K10
Milestones
Received: 31 May 1979
Published: 1 January 1981
Authors
Edward Barry Saff
Richard Steven Varga