  Criteria for the stability of finite sections of a large class of convolution
type operators on $L^p(\mathbb{R})$ are obtained. In this class almost all
classical symbols are permitted, namely operators of multiplication with
functions in $[\textrm{PC} ,\textrm{SO}, L^\infty_0]$ and convolution operators
(as well as Wiener-Hopf and Hankel operators) with symbols in
$[\textrm{PC},\textrm{SO},\textrm{AP},\textrm{BUC}]_p$. We use a simpler and
more powerful algebraic technique than all previous works: the application of
$\mathcal{P}$-theory together with the rich sequences concept and localization.
Beyond stability we study Fredholm theory in sequence algebras. In particular,
formulas for the asymptotic behavior of approximation numbers and Fredholm
indices are given.
