TY - JOUR

T1 - An upper bound for the norm of the Chebyshev polynomial on two intervals

AU - Schiefermayr, Klaus

N1 - Publisher Copyright:
© 2016 Elsevier Inc.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Let E:=[−1,α]∪[β,1], −1<α<β<1, be the union of two real intervals and consider the Chebyshev polynomial of degree n on E, that is, that monic polynomial which is minimal with respect to the supremum norm on E. For its norm, called the n-th Chebyshev number of E, an upper bound in terms of elementary functions of α and β is given. The proof is based on results of N.I. Achieser in the 1930s in which the norm is estimated with the help of Zolotarev's transformation using Jacobi's elliptic and theta functions.

AB - Let E:=[−1,α]∪[β,1], −1<α<β<1, be the union of two real intervals and consider the Chebyshev polynomial of degree n on E, that is, that monic polynomial which is minimal with respect to the supremum norm on E. For its norm, called the n-th Chebyshev number of E, an upper bound in terms of elementary functions of α and β is given. The proof is based on results of N.I. Achieser in the 1930s in which the norm is estimated with the help of Zolotarev's transformation using Jacobi's elliptic and theta functions.

KW - Chebyshev number

KW - Chebyshev polynomial

KW - Jacobi's elliptic functions

KW - Jacobi's theta functions

KW - Logarithmic capacity

KW - Two intervals

UR - http://www.scopus.com/inward/record.url?scp=84984832224&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2016.08.021

DO - 10.1016/j.jmaa.2016.08.021

M3 - Article

SN - 0022-247X

VL - 445

SP - 871

EP - 883

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -