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Inequalities connected with trigonometric sums. (English) Zbl 0754.26008

Constantin Carathéodory: an international tribute. Vol. II, 875-941 (1991).
[For the entire collection see Zbl 0728.00004.]
This is a survey paper on various inequalities for trigonometric polynomials and sums. The main focus is on positivity and monotonicity, although a few other inequalities are included. The importance of positive trigonometric polnomials starts with Fejér’s convergence of the \((C,1)\) means by use of a specific positive trigonometric polynomial. Fejér continued to use and discover positive trigonometric polynomials, with his last paper on them more than fifty years after his first. Through his work his fellow Hungarians, G. Szegö and P. Turán, and others were led to find further useful positive trigonometric polynomials. Fejér had earlier considered a couple of positive polynomials given as sums of Legendre polynomials, and it was his work and related work of Szegö which led to the study of positive sums of Jacobi polynomials, and eventually to some very deep results of Gasper which are waiting for the right uses, as a special case of his joint work with the reviewer was used by de Branges as the final step in his solution of the Bieberbach conjecture.
A few details are given and more than 100 references are included.
Reviewer: R.Askey (Madison)

MSC:

26D05 Inequalities for trigonometric functions and polynomials
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42A05 Trigonometric polynomials, inequalities, extremal problems
26D15 Inequalities for sums, series and integrals

Biographic References:

Carathéodory, C.

Citations:

Zbl 0728.00004
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