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.

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High precision computation of a constant in the theory of trigonometric series

Professor Yudell L. Luke: From tables to algorithms

Let and be sequences of real numbers and let be defined by It is proved that the estimate holds for each natural number such that for all and . Here is the unique root of the equation It is proved that the order in this estimate cannot be improved. Various generalizations of this result are also obtained.Bibliography: 10 titles.

On upper estimates of the partial sums of a trigonometric series in terms of lower estimates

If the partial sums of a trigonometric series are non-negative and two additional conditions are satisfied, then the given series is a Fourier series. One of these conditions is analysed here and necessary expansions and numerical values are given.

Trigonometric sums and fresnel-type integrals

The note "A constant in the theory of trigonometric series" in the October 1964 issue of Mathematics of Computation provided us with a test for our recently constructed algorithms for the computation of roots of functions, and for numerical quadrature in the presence of singularities. The latter algorithm, utilizing the Gaussian 8-point quadrature formula applied to sub-intervals of variable length, involves a sufficiently small number of ordinates that computational labor and round-off error do not become problems. Use of these algorithms indicated the value ao = .3084438, for the root of the equation fJot2 u cos u du = 0, differing from the reported value, .30483, in the third place. To check this result, we made the transformation u = X4 to weaken the character of the singularity at the origin, and obtained the following table by conventional numerical quadrature, confirming our result:

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Geraldine Coombs

Papers

On a constant in the theory of trigonometric series