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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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


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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • Journal: Math. Comp. 39 (1982), 599-615
  • MSC: Primary 30E10; Secondary 65D20
  • DOI:
  • MathSciNet review: 669652