A Lower Bound for the Norm of the Minimal Residual Polynomial

Klaus Schiefermayr

doi:10.1007/s00365-010-9119-2

A Lower Bound for the Norm of the Minimal Residual Polynomial

Klaus Schiefermayr

Research Center Wels

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

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 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.

Original language	English
Pages (from-to)	425-432
Number of pages	8
Journal	Constructive Approximation
Volume	33
Issue number	3
DOIs	https://doi.org/10.1007/s00365-010-9119-2
Publication status	Published - Jun 2011

Keywords

Bernstein-Walsh lemma
Estimated asymptotic convergence factor
Inequality
Inverse polynomial image
Minimal residual polynomial
Minimum deviation

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10.1007/s00365-010-9119-2

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abstract = "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.",

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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.

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KW - Estimated asymptotic convergence factor

KW - Inequality

KW - Inverse polynomial image

KW - Minimal residual polynomial

KW - Minimum deviation

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SP - 425

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JO - Constructive Approximation

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A Lower Bound for the Norm of the Minimal Residual Polynomial

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Keywords

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