Function spaces and operator theory

Abstract

In the mathematics community, wavelets emerged as a refinement of classical Littlewood-Paley methods. Stromberg’s proof that specific spline-type wavelets form an unconditional basis for the real Hardy space ReH¹(ℝⁿ) (see [336]) was a major step in showing that wavelets might also lead to genuinely new mathematical results, although the existence of an explicit unconditional basis—just not exactly wavelets—had already been proved by Carleson [68]. Some feeling emerged that, as a mathematical tool, wavelets might add nothing more than technical simplification of known consequences of Littlewood-Paley theory. Tchamitchian’s construction [342] of certain algebras of singular integrals that are not spectrally invariant finally marked a new achievement in operator theory truly attributable to the use of wavelets. It is not our goal to list specific theorems directly attributable to wavelets, though. It is more enlightening to consider new perspectives that have emerged in response to curmudgeonly challenges.