  In this paper we develop a general theory of compressed sensing for analog
signals, in close similarity to prior results for vectors in finite dimensional
spaces that are sparse in a given orthonormal basis. The signals are modeled by
functions in a reproducing kernel Hilbert space. Sparsity is defined as the
minimal number of terms in expansions based on the kernel functions. Minimizing
this number is under certain conditions equivalent to minimizing an atomic
norm, the pre-dual of the supremum norm for functions in the Hilbert space. The
norm minimizer is shown to exist based on a compactness argument. Recovery
based on minimizing the atomic norm is robust and stable, so it provides
controllable accuracy for recovery when the signal is only approximately sparse
and the measurement is corrupted by noise.
  As applications of the theory, we include results on the recovery of sparse
bandlimited functions and functions that have a sparse inverse short-time
Fourier transform.
