High precision computation of a constant in the theory of trigonometric series. (English) Zbl 1202.42003
Summary: 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.
MSC:
| 42-04 | Software, source code, etc. for problems pertaining to harmonic analysis on Euclidean spaces |
| 42A05 | Trigonometric polynomials, inequalities, extremal problems |
| 11Y60 | Evaluation of number-theoretic constants |
| 65H05 | Numerical computation of solutions to single equations |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 42A10 | Trigonometric approximation |
References:
| [1] | Richard Askey and John Steinig, Some positive trigonometric sums, Trans. Amer. Math. Soc. 187 (1974), 295 – 307. · Zbl 0244.42002 |
| [2] | Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. · Zbl 0298.33008 |
| [3] | 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. · Zbl 0797.33006 · doi:10.1016/0377-0427(93)90312-Y |
| [4] | 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). |
| [5] | R. P. Boas Jr. and Virginia C. Klema, A constant in the theory of trigonometric series, Math. Comp. 18 (1964), 674. |
| [6] | 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. · Zbl 0792.42002 · doi:10.1017/S030500410007167X |
| [7] | Gavin Brown, Feng Dai, and Kunyang Wang, On positive cosine sums, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 2, 219 – 232. · Zbl 1148.42003 · doi:10.1017/S0305004106009947 |
| [8] | R. F. Church, On a constant in the theory of trigonometric series, Math. Comp. 19 (1965), 501. · Zbl 0134.33101 |
| [9] | Steven R. Finch, Mathematical constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Cambridge, 2003. · Zbl 1054.00001 |
| [10] | 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). · JFM 55.0753.02 · doi:10.1007/BF02410076 |
| [11] | J. Keane, Estimating Brown-Wang \( B\) and Zygmund \( R\) constants, unpublished note (2000). |
| [12] | Stamatis Koumandos and Stephan Ruscheweyh, Positive Gegenbauer polynomial sums and applications to starlike functions, Constr. Approx. 23 (2006), no. 2, 197 – 210. · Zbl 1099.30006 · doi:10.1007/s00365-004-0584-3 |
| [13] | Stamatis Koumandos and Stephan Ruscheweyh, On a conjecture for trigonometric sums and starlike functions, J. Approx. Theory 149 (2007), no. 1, 42 – 58. · Zbl 1135.42001 · doi:10.1016/j.jat.2007.04.006 |
| [14] | 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. · Zbl 0134.33102 |
| [15] | A. Zygmund, Trigonometric series. Vol. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition. · Zbl 0628.42001 |
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