  Recently, it was discovered that for a given function class $\mathbf{F}$ the
error of best linear recovery in the square norm can be bounded above by the
Kolmogorov width of $\mathbf{F}$ in the uniform norm. That analysis is based on
deep results in discretization of the square norm of functions from finite
dimensional subspaces. In this paper we show how very recent results on
universal discretization of the square norm of functions from a collection of
finite dimensional subspaces lead to an inequality between optimal sparse
recovery in the square norm and best sparse approximations in the uniform norm
with respect to appropriate dictionaries.
