  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.
