  We consider the noncommutative space $\mathbb{R}^3_\lambda$, a deformation of
the algebra of functions on $\mathbb{R}^3$ which yields a "foliation" of
$\mathbb{R}^3$ into fuzzy spheres. We first construct a natural matrix base
adapted to $\mathbb{R}^3_\lambda$. We then apply this general framework to the
one-loop study of a two-parameter family of real-valued scalar noncommutative
field theories with quartic polynomial interaction, which becomes a non-local
matrix model when expressed in the above matrix base. The kinetic operator
involves a part related to dynamics on the fuzzy sphere supplemented by a term
reproducing radial dynamics. We then compute the planar and non-planar 1-loop
contributions to the 2-point correlation function. We find that these diagrams
are both finite in the matrix base. We find no singularity of IR type, which
signals very likely the absence of UV/IR mixing. We also consider the case of a
kinetic operator with only the radial part. We find that the resulting theory
is finite to all orders in perturbation expansion.
