Mathematics – Classical Analysis and ODEs
Scientific paper
2008-09-23
Transactions of the American Mathematical Society, v. 362, p. 5803-5803, 2010
Mathematics
Classical Analysis and ODEs
40 pages. Accepted for publication in Trans. Amer. Math. Soc
Scientific paper
10.1090/S0002-9947-2010-04886-X
We obtain extremal majorants and minorants of exponential type for a class of even functions on $\R$ 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 Erd\"{o}s-Tur\'{a}n-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.
Carneiro Emanuel
Vaaler Jeffrey D.
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