  We consider estimation in a sparse additive regression model with the design
points on a regular lattice. We establish the minimax convergence rates over
Sobolev classes and propose a Fourier-based rate-optimal estimator which is
adaptive to the unknown sparsity and smoothness of the response function. The
estimator is derived within Bayesian formalism but can be naturally viewed as a
penalized maximum likelihood estimator with the complexity penalties on the
number of nonzero univariate additive components of the response and on the
numbers of the nonzero coefficients of their Fourer expansions. We compare it
with several existing counterparts and perform a short simulation study to
demonstrate its performance.
