A remark on the representation of Zolotarev polynomials

Zinoviy Grinshpun

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Zolotarev polynomials are the polynomials that have minimal deviation from zero on [-1, 1] with respect to the norm xn - σ xn-1 + an-2 xn-2 + ... + a1x + an for given σ and for all ak ∈ ℛ.This note complements the paper of F. Pehersforfer [J. London Math. Soc. (1) 74 (2006) 143-153] with exact (not asymptotic) construction of the Zolotarev polynomials with respect to the norm L1 for σ < 1 and with respect to the norm L2 for σ ≠ 1 in the form of Bernstein-Szegö orthogonal polynomials. For all σ ∈ ℛ in L1 and L2 norms, the Zolotarev polynomials satisfy exactly (not asymptotically) the triple recurrence relation of the Chebyshev polynomials.

Original languageEnglish
Pages (from-to)139-142
Number of pages4
JournalBulletin of the London Mathematical Society
Issue number1
StatePublished - Feb 2008


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