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

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• 作者：Klaus Schiefermayr ^{klaus.schiefermayr@fh-wels.at" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Chebyshev number ; Chebyshev polynomial ; Jacobi's elliptic functions ; Jacobi's theta functions ; Logarithmic capacity ; Two intervals
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：871-883
• 全文大小：351 K

文摘

Let E:=[−1,α]∪[β,1]$E:=[-1,\alpha ]\cup [\beta ,1]$, −1<α<β<1$-1<\alpha <\beta <1$, 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.