Approximation sequences on Banach spaces: a rich approach

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.