A Lower Bound for the Norm of the Minimal Residual Polynomial

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5 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)425-432
Number of pages8
JournalConstructive Approximation
Issue number3
Publication statusPublished - Jun 2011


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


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