We define new symbol classes for pseudodifferntial operators and investigate
their pseudodifferential calculus. The symbol classes are parametrized by
commutative convolution algebras. To every solid convolution algebra over a
lattice we associate a symbol class. Then every operator with such a symbol is
almost diagonal with respect to special wave packets (coherent states or Gabor
frames), and the rate of almost diagonalization is described precisely by the
underlying convolution algebra. Furthermore, the corresponding class of
pseudodifferential operators is a Banach algebra of bounded operators on L2. If a version of Wiener's lemma holds for the underlying convolution algebra,
then the algebra of pseudodifferential operators is closed under inversion. The
theory contains as a special case the fundamental results about Sj\"ostrand's
class and yields a new proof of a theorem of Beals about the H\"ormander class
of order 0.Comment: 28 page