High precision computation of a constant in the theory of trigonometric series
HTML articles powered by AMS MathViewer
- by J. Arias de Reyna and J. van de Lune PDF
- Math. Comp. 78 (2009), 2187-2191 Request permission
Abstract:
Using the bisection as well as the Newton-Raphson method, we compute to high precision the Littlewood-Salem-Izumi constant frequently occurring in the theory of trigonometric sums.References
- Richard Askey and John Steinig, Some positive trigonometric sums, Trans. Amer. Math. Soc. 187 (1974), 295–307. MR 338481, DOI 10.1090/S0002-9947-1974-0338481-3
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
- Richard Askey, Problems which interest and/or annoy me, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 3–15. MR 1246848, DOI 10.1016/0377-0427(93)90312-Y
- A. S. Belov, Coefficients of trigonometric cosine series with nonnegative partial sums, Trudy Mat. Inst. Steklov. 190 (1989), 3–21 (Russian). Translated in Proc. Steklov Inst. Math. 1992, no. 1, 1–18; Theory of functions (Russian) (Amberd, 1987). MR 1005335
- R. P. Boas Jr. and Virginia C. Klema, A constant in the theory of trigonometric series, Math. Comp. 18 (1964), 674. MR 176283, DOI 10.1090/S0025-5718-1964-0176283-9
- Gavin Brown, Kun Yang Wang, and David C. Wilson, Positivity of some basic cosine sums, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 3, 383–391. MR 1235986, DOI 10.1017/S030500410007167X
- Gavin Brown, Feng Dai, and Kunyang Wang, On positive cosine sums, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 2, 219–232. MR 2314596, DOI 10.1017/S0305004106009947
- R. F. Church, On a constant in the theory of trigonometric series, Math. Comp. 19 (1965), 501.
- Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Cambridge, 2003. MR 2003519
- Karl Grandjot, Vojtěch Jarnik, Edmund Landau, and John Edensor Littlewood, Bestimmung einer absoluten Konstanten aus der Theorie der trigonometrischen Reihen, Ann. Mat. Pura Appl. 6 (1979), no. 1, 1–7 (German). MR 1553122, DOI 10.1007/BF02410076
- J. Keane, Estimating Brown-Wang $B$ and Zygmund $R$ constants, unpublished note (2000).
- Stamatis Koumandos and Stephan Ruscheweyh, Positive Gegenbauer polynomial sums and applications to starlike functions, Constr. Approx. 23 (2006), no. 2, 197–210. MR 2186305, DOI 10.1007/s00365-004-0584-3
- Stamatis Koumandos and Stephan Ruscheweyh, On a conjecture for trigonometric sums and starlike functions, J. Approx. Theory 149 (2007), no. 1, 42–58. MR 2371613, DOI 10.1016/j.jat.2007.04.006
- Y. L. Luke, W. Fair, G. Coombs and R. Moran, On a constant in the theory of trigonometric series, Math. Comp. 19 (1965), 501–502.
- A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. MR 933759
Additional Information
- J. Arias de Reyna
- Affiliation: Facultad de Matemáticas, Universidad de Sevilla, Apdo. 1160, 41080-Sevilla, Spain
- ORCID: 0000-0003-3348-4374
- Email: arias@us.es
- J. van de Lune
- Affiliation: Langebuorren 49, 9074 CH Hallum, The Netherlands \text{(formerly at CWI, Amsterdam)}
- Email: j.vandelune@hccnet.nl
- Received by editor(s): July 28, 2008
- Received by editor(s) in revised form: September 21, 2008
- Published electronically: February 9, 2009
- Additional Notes: The first author was supported by MCI Grant MTM2006-05622.
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 78 (2009), 2187-2191
- MSC (2000): Primary 42-04, 26D05; Secondary 11Y60
- DOI: https://doi.org/10.1090/S0025-5718-09-02222-4
- MathSciNet review: 2521284