TOPICS
Search

Littlewood-Salem-Izumi Constant


Zygmund (1988, p. 192) noted that there exists a number alpha_0 in (0,1) such that for each alpha>=alpha_0, the partial sums of the series sum_(n=1)^(infty)n^(-alpha)cos(nx) are uniformly bounded below, whereas for alpha<=alpha_0, they are not (Arias de Reyna and van de Lune 2009).

This constant is given by the unique solution for 0<alpha<1 of

int_0^(3pi/2)u^(-alpha)cosudu=(1-alpha)^(-1)((3pi)/2)^(1-alpha)_1F_2(1/2(1-alpha);1/2,1/2(3-alpha);-9/(16)pi^2)
(1)
=0,
(2)

where _1F_2(a;b,c;z) is a generalized hypergeometric function, which is given by alpha_0=0.3084437795... (OEIS A157957).

The origin of the defining property for alpha_0 appeared in an unpublished result of Littlewood and Salem and the equation defining alpha_0 is due to S. Izumi (Zygmund 1988, p. 379), thus justifying the name Littlewood-Salem-Izumi constant (Arias de Reyna and van de Lune 2009).


Explore with Wolfram|Alpha

References

Arias de Reyna, J. and van de Lune, J. "High Precision Computation of a Constant in the Theory of Trigonometric Series." Math. Comput. Published electronically, February 9, 2009.Askey, R. Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM, 1975.Belov, A. S. "Coefficients of Trigonometric Cosine Series with Nonnegative Partial Sums." Translated in Proc. Steklov Inst. Math. 1992, 1-18, 1992. "Theory of Functions. (Amberd, 1987)." Trudy Mat. Inst. Steklov, 190, pp. 3-21, 1989.Boas, R. P. Jr. and Klema, C. "A Constant in the Theory of Trigonometric Series." Math. Comput. 18, 674, 1964.Brown, G.; Wang, K.; and Wilson, D. C. "Positivity of Some Basic Cosine Sums." Math. Proc. Cambridge Philos. Soc. 114, 383-391, 1993.Brown, G.; Dai, F.; and Wang, K. "On Positive Cosine Sums." Math. Proc. Cambridge Philos. Soc. 142, 219-232, 2007.Church, R. F. "On a Constant in the Theory of Trigonometric Series." Math. Comput. 19, 501, 1965.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Luke, Y. L.; Fair, W.; Coombs, G.; and Moran, R. "On a Constant in the Theory of Trigonometric Series." Math. Comput. 19, 501-502, 1965.Grandjot, K.; Jarnik, V.; Landau, E.; and Littlewood, J. E. "Bestimmung einer absoluten Konstanten aus der Theorie der trigonometrischen Reihen." Annali di Mat. 6, 1-7, 1929.Koumandos, S. and Ruscheweyh, S. "Positive Gegenbauer Polynomial Sums and Applications to Starlike Functions." Constr. Approx. 23, 197-210, 2006.Sloane, N. J. A. Sequence A157957 in "The On-Line Encyclopedia of Integer Sequences."Zygmund, A. G. Trigonometric Series, Vols. 1-2, 2nd ed. New York: Cambridge University Press, 1988.

Referenced on Wolfram|Alpha

Littlewood-Salem-Izumi Constant

Cite this as:

Weisstein, Eric W. "Littlewood-Salem-Izumi Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Littlewood-Salem-IzumiConstant.html

Subject classifications