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Complex Chebyshev polynomials on circular sectors with degree six or less

Authors: U. Grothkopf and G. Opfer
Journal: Math. Comp. 39 (1982), 599-615
MSC: Primary 30E10; Secondary 65D20
MathSciNet review: 669652
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Abstract: Let $ T_n^\alpha $ denote the nth Chebyshev polynomial on the circular sector $ {S^\alpha } = \{ z:\vert z\vert \leqslant 1,\vert\arg z\vert \leqslant \alpha \} $. This paper contains numerical values of $ {\left\Vert {T_n^\alpha } \right\Vert _\infty }$ and the corresponding coefficients of $ T_n^\alpha $ for $ n = 1(1)6$ and $ \alpha = {0^ \circ }({5^ \circ }){180^ \circ }$. Also all critical angles for $ T_n^\alpha ,n = 1(1)6$ are listed, where an angle is called critical when the number of absolute maxima of $ \vert T_n^\alpha \vert$ changes at that angle. All figures are given to six places. The positions (and hence the number) of extremal points of $ T_n^\alpha ,n = 1(1)6$ are presented graphically. The method consists of a combination of semi-infinite linear programming, finite linear programming, and Newton's method.

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Article copyright: © Copyright 1982 American Mathematical Society

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