Summary: The space in question is \(A_ \mu({\mathbf R}):=L_ 1({\mathbf R})+B_ \mu({\mathbf R})\), where \(B_ \mu({\mathbf R})\) is a Banach space that contains the “tails” (the dominant parts for large values of \(| x|)\) of certain slowly decreasing functions from \({\mathbf R}\) to \({\mathbf R}\). Functions in \(B_ \mu({\mathbf R})\) are of bounded variation, and the norm involves their variation and a weighting function. Theorems are proved only for \(B_ \mu({\mathbf R})\), because those for \(L_ 1({\mathbf R})\) are known. The results concern the convolution of a function in \(B_ \mu({\mathbf R})\) with one in \(L_ 1({\mathbf R})\), the Fourier transform acting on \(B_ \mu({\mathbf R})\), and the signum rule for the Hilbert transform of functions in \(B_ \mu({\mathbf R})\).