CHAPTER II - FOURIER TRANSFORMS OF GENERALIZED FUNCTIONS

Abstract

This chapter focuses on Fourier transforms of generalized functions. It explores Fourier transforms of functions in K. The study of the Fourier transforms of functions in K leads naturally to define the new space Z of slowly increasing functions, namely, of all entire functions ψ(s) satisfying the inequalities. Fourier transform establishes a one-to-one mapping between K and Z. This mapping preserves the linear operations. This implies that every linear operator defined on K there corresponds a dual operator defined on Z. The chapter discusses Fourier transforms of functions in S. There exists a one-to-one mapping between K and Z that preserves the linear operations and convergence. A similar mapping can be established between the continuous linear functionals on these spaces. This mapping is set up so that when applied to a regular functional corresponding to an absolutely integrable function, it induces a mapping of this function into its classical Fourier transform. The Fourier transforms of periodic functions, generalized functions with bounded support, all ordinary functions increasing at infinity no faster than some power of their argument, and the Fourier transforms of their derivatives can be considered functionals on S.