Quasi-banded operators, convolutions with almost periodic or quasi-continuous data, and their approximations

We study the stability and Fredholm property of the finite sections of quasi-banded operators acting on L p spaces over the real line. This family is significantly larger than the set of band-dominated operators, but still permits to derive criteria for the stability and results on the splitting property, as well as an index formula in the form as it is known for the classical cases. In particular, this class covers convolution type operators with semi-almost periodic and quasi-continuous symbols, and operators of multiplication by slowly oscillating, almost periodic or even more general coefficients.