A Sequence Algebra of Finite Sections, Convolution, and Multiplication Operators on LP()

This paper is concerned with the applicability of the finite sections method to operators belonging to the closed subalgebra of L(Lp()), 1p , generated by operators of multiplication by piecewise continuous functions in [image omitted] 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.