The Lebesgue Constants for Regular Hausdorff Methods

Abstract

The unboundedness of the sequence of Lebesgue constants (norms), at a point, of certain transforms implies, as is well known, that there exist (i) a continuous function whose transform fails to converge to the function at the point in question (the du Bois-Reymond singularity), and (ii) another such function whose transform, while converging everywhere to the function, does not do so uniformly in any neighbourhood of the stipulated point (the Lebesgue singularity). The converses also hold in our case.