  We study the performance of adaptive Fourier-Galerkin methods in a periodic
box in $\mathbb{R}^d$ with dimension $d\ge 1$. These methods offer unlimited
approximation power only restricted by solution and data regularity. They are
of intrinsic interest but are also a first step towards understanding
adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes,
one classical corresponding to algebraic decay of Fourier coefficients and
another associated with exponential decay. We study the sparsity classes of the
residual and show that they are the same as the solution for the algebraic
class but not for the exponential one. This possible sparsity degradation for
the exponential class can be compensated with coarsening, which we discuss in
detail. We present several adaptive Fourier algorithms, and prove their
contraction and optimal cardinality properties.
