A lower bound for the norm of the minimal residual polynomial

2013/06/25 by Klaus Schiefermayr, Schiefermayr, Klaus · 1 citation
Mathematics · #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #math.CA #math.CV

paper · pdf · doi:10.48550/arxiv.1306.5868

arxiv created 2013/06/25 · openalex publication_date 2013/06/25 · arxiv updated 2013/06/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 κ(S):=lim_n→∞√[n]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.

Citations

• Convergence Analysis of Krylov Subspace Iterations with Methods from Potential Theory
• Extremal polynomials associated with a system of curves in the complex plane
• A lower bound for the minimum deviation of the Chebyshev polynomial on a compact real set

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• Unbounded Widom factors for orthogonal and residual polynomials

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