Helena Mascarenhas

Helena Mascarenhas, Pedro A. Santos, Markus Seidel

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.

Alexei Yu. Karlovich, Helena Mascarenhas, Pedro A. Santos

We prove the applicability of the finite section method to an arbitrary operator in the Banach algebra generated by the operators of multiplication by piecewise continuous functions and the convolution operators with symbols in the algebra generated by piecewise continuous and slowly oscillating Fourier multipliers on $L^p(\mathbb{R})$, $1<p<\infty$.