Some extremal functions in Fourier analysis.
by
Emanuel Carneiro
University of Texas at Austin
Coauthors: Jeffrey Vaaler - University of Texas at Austin

We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log|x| and |x|^a, where -1 < a < 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 on the roots of the polynomials.

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Date received: January 29, 2008