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Mathematics of Computation

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ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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Complex Chebyshev polynomials on circular sectors with degree six or less
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by U. Grothkopf and G. Opfer PDF
Math. Comp. 39 (1982), 599-615 Request permission

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.
References
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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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