TY - JOUR
T1 - A Lower Bound for the Norm of the Minimal Residual Polynomial
AU - Schiefermayr, Klaus
N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/6
Y1 - 2011/6
N2 - Let S be a compact infinite set in the complex plane with 0 ∉ S, and let Rn 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 Rn(0)=1. For the norm Ln(S) of the minimal residual polynomial, the limit exists. In addition to the well-known and widely referenced inequality Ln(S) ≥ κ(S)n, we derive the sharper inequality Ln(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.
AB - Let S be a compact infinite set in the complex plane with 0 ∉ S, and let Rn 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 Rn(0)=1. For the norm Ln(S) of the minimal residual polynomial, the limit exists. In addition to the well-known and widely referenced inequality Ln(S) ≥ κ(S)n, we derive the sharper inequality Ln(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.
KW - Bernstein-Walsh lemma
KW - Estimated asymptotic convergence factor
KW - Inequality
KW - Inverse polynomial image
KW - Minimal residual polynomial
KW - Minimum deviation
UR - http://www.scopus.com/inward/record.url?scp=79952987254&partnerID=8YFLogxK
U2 - 10.1007/s00365-010-9119-2
DO - 10.1007/s00365-010-9119-2
M3 - Article
SN - 0176-4276
VL - 33
SP - 425
EP - 432
JO - Constructive Approximation
JF - Constructive Approximation
IS - 3
ER -