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An Extremal Problem Concerning the Derivatives of a Polynomial at Its Roots
Author(s) -
OaxacaAdams Guillermo,
VerdeStar Luis
Publication year - 1997
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/1467-9590.00040
Subject(s) - mathematics , vandermonde matrix , monic polynomial , polynomial , interval (graph theory) , combinatorics , properties of polynomial roots , inverse , infinity , degree (music) , inverse problem , discrete mathematics , matrix polynomial , mathematical analysis , eigenvalues and eigenvectors , geometry , physics , quantum mechanics , acoustics
Let w be a monic polynomial of degree n +1 with roots x j in the interval [−1, 1]. We consider the problem of finding the roots x j for which the minimum of ¦ w ′( x j )¦, for 0≤ j ≤ n , is as large as possible. We prove that the Clenshaw–Curtis points cos( j π/ n ) are the only solution when n is even and that they get asymptotically close to the solution for odd values of n , as n goes to infinity. Our problem is related to the problem of minimizing the norm of inverse Vandermonde matrices.