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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
DOI: https://doi.org/10.1090/S0025-5718-1982-0669652-2
MathSciNet review: 669652
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Abstract: Let $T_n^\alpha$ denote the nth Chebyshev polynomial on the circular sector ${S^\alpha } = \{ z:|z| \leqslant 1,|\arg z| \leqslant \alpha \}$. This paper contains numerical values of ${\left \| {T_n^\alpha } \right \|_\infty }$ 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 $|T_n^\alpha |$ 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