A uniqueness theorem for square integrable high-pass functions

Summary

The principal aim of this paper is to indicate that the theory of α and β functions, the boundary functions of analytic functions in the upper and lower half plane belonging to the Hardy classH ₂, unexpectedly reveals a close connection with the theory of high-pass functions, i.e. the Fouriertransforms of functions vanishing on (−a, a), as given by Logan. In several papers, I have developed the subject of α and β functions carried by the fact that they represent relatively easy and thus fascinating examples of classes of quasi-analytic functions on the real axis (1). It happens, in fact, that the basic uniqueness theorem on high-pass functions becomes considerably simpler and transparent by the use of the structure inherent to boundary functions of analytic functions, such as α or β functions. For comparison, a proof is first given below which works out the path indicated by Logan and becomes somewhat easier by the fact that here we deal with theL ₂ case only (2).