  We construct majorants and minorants of a Gaussian function in Euclidean
space that have Fourier transforms supported in a box. The majorants that we
construct are shown to be extremal and our minorants are shown to be
asymptotically extremal as the sides of the box become uniformly large. We then
adapt the Distribution and Gaussian Subordination methods of
Carneiro-Littmann-Vaaler to the multidimensional setting to obtain majorants
and minorants for a class of radial functions. Periodic analogues of the main
results are proven and applications to Hilbert-type inequalities are given.
