Some extremal functions in Fourier analysis. II

Carneiro, Emanuel; Vaaler, Jeffrey

doi:10.1090/S0002-9947-2010-04886-X

Some extremal functions in Fourier analysis. II
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by Emanuel Carneiro and Jeffrey D. Vaaler PDF

Trans. Amer. Math. Soc. 362 (2010), 5803-5843 Request permission

Abstract:

We obtain extremal majorants and minorants of exponential type for a class of even functions on $\mathbb {R}$ which includes $\log |x|$ and $|x|^\alpha$, where $-1 < \alpha < 1$. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms, which include discrete analogues of the one dimensional Hardy-Littlewood-Sobolev inequalities. A further application provides an Erdös-Turán-type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials.

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