Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz & I. A. Stegun (Editors), Handbook of Mathematical Functions, fifth printing, Dover, New York, 1968.
  • [2] J. P. Coleman & A. J. Monaghan, Chebyshev Expansions for the Bessel Function $ {J_n}(z)$ in the Complex Plane, University of Durham, 1980. (Preprint.)
  • [3] G. H. Elliott, Polynomial Approximation in the Complex Plane Using Generalised Humbert Polynomials, Lecture at Dundee Biennial Conference on Numerical Analysis, 1981.
  • [4] C. Geiger & G. Opfer, "Complex Chebyshev polynomials on circular sectors," J. Approx. Theory, v. 24, 1978, pp. 93-118. MR 511466 (80m:41004)
  • [5] K. Glashoff & S. Å. Gustafson, Einführung in die lineare Optimierung, Wissenschaftliche Buchgesellschaft, Darmstadt, 1978.
  • [6] K. Glashoff & K. Roleff, "A new method for Chebyshev approximation of complex-valued functions," Math. Comp., v. 36, 1981, pp. 233-239. MR 595055 (82c:65011)
  • [7] R. Hettich, "Numerical methods for nonlinear Chebyshev approximation," in Approximation in Theorie und Praxis (G. Meinardus, Ed.), Bibliographisches Institut Mannheim, Wien, Zürich, 1979, pp. 139-156. MR 567658 (81g:65021)
  • [8] G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer-Verlag, Berlin, Heidelberg and New York, 1967. MR 0217482 (36:571)
  • [9] G. Opfer, "An algorithm for the construction of best approximations based on Kolmogorov's criterion," J. Approx. Theory, v. 23, 1978, pp. 299-317. MR 509560 (80i:65019)
  • [10] R. L. Streit & A. H. Nuttall, Linear Chebyshev Complex Function Approximation, Technical Report 6403, Naval Underwater Systems Center, Newport, R. I., New London, Conn., 1981.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 30E10, 65D20

Retrieve articles in all journals with MSC: 30E10, 65D20


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1982-0669652-2
Article copyright: © Copyright 1982 American Mathematical Society

American Mathematical Society