Algebra of convolution type operators with continuous data on Banach function spaces

We show that if the Hardy–Littlewood maximal operator is bounded on a reflexive Banach function space X(R) and on its associate space X′(R), then the space X(R) has an unconditional wavelet basis. As a consequence of the existence of a Schauder basis in X(R), we prove that the ideal of compact operators K(X(R)) on the space X(R) is contained in the Banach algebra generated by all operators of multiplication aI by functions a∈C(R˙), where R˙=R∪{∞}, and by all Fourier convolution operators W0(b) with symbols b∈CX(R˙), the Fourier multiplier analogue of C(R˙).