A Lower Bound for the Norm of the Minimal Residual Polynomial

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...

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Bibliographic Details
Published in:Constructive approximation 2011-06, Vol.33 (3), p.425-432
Main Author: Schiefermayr, Klaus
Format: Article
Language:English
Subjects:
Analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
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Summary: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 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 )≥2 κ ( S ) n /(1+ κ ( S ) 2 n ) 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.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-010-9119-2