  We investigate two systematic constructions of inverse-closed subalgebras of
a given Banach algebra or operator algebra A, both of which are inspired by
classical principles of approximation theory. The first construction requires a
closed derivation or a commutative automorphism group on A and yields a family
of smooth inverse-closed subalgebras of A that resemble the usual
Holder-Zygmund spaces. The second construction starts with a graded sequence of
subspaces of A and yields a class of inverse-closed subalgebras that resemble
the classical approximation spaces. We prove a theorem of Jackson-Bernstein
type to show that in certain cases both constructions are equivalent.
  These results about abstract Banach algebras are applied to algebras of
infinite matrices with off-diagonal decay. In particular, we obtain new and
unexpected conditions of off-diagonal decay that are preserved under matrix
inversion.
