  This paper presents the Gaussian subordination framework to generate optimal
one-sided approximations to multidimensional real-valued functions by functions
of prescribed exponential type. Such extremal problems date back to the works
of Beurling and Selberg and provide a variety of applications in analysis and
analytic number theory. Here we majorize and minorize (on $\mathbb{R}^N$) the
Gaussian ${\bf x} \mapsto e^{-\pi \lambda |{\bf x}|^2}$, where $\lambda >0$ is
a free parameter, by functions with distributional Fourier transforms supported
on Euclidean balls, optimizing weighted $L^1$-errors. By integrating the
parameter $\lambda$ against suitable measures, we solve the analogous problem
for a wide class of radial functions. Applications to inequalities and periodic
analogues are discussed. The constructions presented here rely on the theory of
de Branges spaces of entire functions and on new interpolations tools derived
from the theory of Laplace transforms of Laguerre-P\'{o}lya functions.
