  Motivated by the question of optimal functional approximation via compressed
sensing, we propose generalizations of the Iterative Hard Thresholding and the
Compressive Sampling Matching Pursuit algorithms able to promote sparse in
levels signals. We show, by means of numerical experiments, that the proposed
algorithms are successfully able to outperform their unstructured variants when
the signal exhibits the sparsity structure of interest. Moreover, in the
context of piecewise smooth function approximation, we numerically demonstrate
that the structure promoting decoders outperform their unstructured variants
and the basis pursuit program when the encoder is structure agnostic.
