Convolution Type Operators on ℝ^N

Abstract

In this chapter, we are going to introduce a Banach algebra B_p of operators onL^p(ℝ^N) which become band-dominated operators acting onlP(ℤ^N,X) withX= L^p([0, 1)^N) after a suitable discretization and which are, thus, subject to the general theory developed in the previous chapter. Prominent members of this class are the Fourier convolution operators, certain pseudodifferential operators as well as operators in finite differences. We employ the close relationship between operators in B_pand their discretizations to study the local invertibility at infinity (which is related with the Fredholmness) as well as the applicability of the finite section method for operators in B_p.