An extension of the gel'fand-shilov technique for hilbert transforms

Abstract

Gel'fand and Shilov in [3; pp. 151-54] have extended the Hilbert transform to generalized functions of the space Φ' and proved the inversion formula where the elements of the testing function space Φ belong to the dual space of a testing function space Ψ equipped with the topology generated by a countable set of norms. They claimed that a locally integrable function f(χ) satisfying the asymptotic order and 0<e<1, is Hilbert transformable according to their theory. As it stands their this claim is incorrect: in this paper we have modified their technique and have extended the Hilbert transform and its inversion formula to the space of tempered distributions on . We have given very simple examples to illustrate the applications of our results in solving some singular differential and integro-differential equations. AMS Subject Classification (1980) Primary 46F12 Secondary 44A15