We determine extremal entire functions for the problem of majorizing,
minorizing, and approximating the Gaussian function e−πλx2 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∣α
for −1<α; \,log((x2+α2)/(x2+β2)), for
0≤α<β;\, log(x2+α2); and x2nlogx2\,, for n∈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