A Sequence Algebra of Finite Sections, Convolution, and Multiplication Operators on L P (ℝ)

Abstract

This paper is concerned with the applicability of the finite sections method to operators belonging to the closed subalgebra of ℒ(L p (ℝ)), 1 < p < ∞, generated by operators of multiplication by piecewise continuous functions in and operators of convolution by piecewise continuous Fourier multipliers. For this, we introduce a larger algebra of sequences, which contains the sequences that interest us and the usual operator algebra generated by the operators of multiplication and convolution. There is a direct relationship between the applicability of the finite sections method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship and using local principles, we construct locally equivalent representations that allow one to derive invertibility criteria.