  We determine extremal entire functions for the problem of majorizing,
minorizing, and approximating the Gaussian function $e^{-\pi\lambda x^2}$ by
entire functions of exponential type. This leads to the solution of analogous
extremal problems for a wide class of even functions that includes most of the
previously known examples (for instance \cite{CV2}, \cite{CV3}, \cite{GV} and
\cite{Lit}), plus a variety of new interesting functions such as $|x|^{\alpha}$
for $-1 < \alpha$; \,$\log \,\bigl((x^2 + \alpha^2)/(x^2 + \beta^2)\bigr)$, for
$0 \leq \alpha < \beta$;\, $\log\bigl(x^2 + \alpha^2\bigr)$; and $x^{2n} \log
x^2$\,, for $n \in \N$. Further applications to number theory include optimal
approximations of theta functions by trigonometric polynomials and optimal
bounds for certain Hilbert-type inequalities related to the discrete
Hardy-Littlewood-Sobolev inequality in dimension one.
