Sparsity for isotropic spherical random fields

We introduce a simple representation for isotropic spherical random fields and we discuss how it allows to discuss different notions of sparsity under isotropy. We also show how a suitable construction of sparse fields can mimic well the angular power spectrum and the polyspectra of some popular non-Gaussian fields, at the same time allowing for computationally efficient simulation algorithms. Using related ideas we also show how it is possible to obtain sparse approximations of spherical random fields which preserve isotropy, thus addressing an issue which has been raised in the Cosmological literature.