A Lower Bound for the Norm of the Minimal Residual Polynomial |
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Authors: | Klaus Schiefermayr |
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Institution: | 1. School of Engineering and Environmental Sciences, Upper Austria University of Applied Sciences, Stelzhamerstr. 23, 4600, Wels, Austria
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Abstract: | Let S be a compact infinite set in the complex plane with 0∉S, and let R
n
be the minimal residual polynomial on S, i.e., the minimal polynomial of degree at most n on S with respect to the supremum norm provided that R
n
(0)=1. For the norm L
n
(S) of the minimal residual polynomial, the limit k(S):=limn?¥n?{Ln(S)}\kappa(S):=\lim_{n\to\infty}\sqrtn]{L_{n}(S)} exists. In addition to the well-known and widely referenced inequality L
n
(S)≥κ(S)
n
, we derive the sharper inequality L
n
(S)≥2κ(S)
n
/(1+κ(S)2n
) in the case that S is the union of a finite number of real intervals. As a consequence, we obtain a slight refinement of the Bernstein–Walsh
lemma. |
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Keywords: | |
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