  The purpose of this paper is to characterize all the entire solutions of the
homogeneous Helmholtz equation (solutions in $\mathbb{R}^d$) arising from the
Fourier extension operator of distributions in Sobolev spaces of the sphere
$H^\alpha(\mathbb{S}^{d-1}),$ with $\alpha\in \mathbb{R}$. We present two
characterizations. The first one is written in terms of certain $L^2$-weighted
norms involving real powers of the spherical Laplacian. The second one is in
the spirit of the classical description of the Herglotz wave functions given by
P. Hartman and C. Wilcox. For $\alpha>0$ this characterization involves a
multivariable square function evaluated in a vector of entire solutions of the
Helmholtz equation, while for $\alpha<0$ it is written in terms of an spherical
integral operator acting as a fractional integration operator. Finally, we also
characterize all the solutions that are the Fourier extension operator of
distributions in the sphere.
