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Bulletin of the London Mathematical Society Advance Access originally published online on February 4, 2008
Bulletin of the London Mathematical Society 2008 40(1):139-142; doi:10.1112/blms/bdm109
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© 2008 London Mathematical Society

A remark on the representation of Zolotarev polynomials

Zinoviy Grinshpun

Department of Mathematics
Bar-Ilan University
Ramat-Gan 52900
Israel

Received 11 February 2007. Revision received 31 July 2007.

Zolotarev polynomials are the polynomials that have minimal deviation from zero on [–1, 1] with respect to the norm ||xn – {sigma} xn–1 + an–2 xn–2 + ... + a1x + an|| for given {sigma} and for all ak isin 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 |{sigma}| < 1 and with respect to the norm L2 for |{sigma}| != 1 in the form of Bernstein–Szegö orthogonal polynomials. For all {sigma} isin R in L1 and L2 norms, the Zolotarev polynomials satisfy exactly (not asymptotically) the triple recurrence relation of the Chebyshev polynomials.

2000 Mathematics Subject Classification 33C45, 42C05, 42C15.


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