A Handbook of Fourier Theorems

Functionals of types D′ and Z′

In order to give meaning to the Fourier transform of any locally integrable function it is necessary to generalize beyond the functionals in S′, and we describe now the functionals in classes D′ and Z′ which provide the necessary generalization. We give examples in section 14.2, but start in this section with the basic concepts.

In summary, the functionals of class D′ are based on the use of test functions in class D (the good functions of bounded support), whilst the functionals in class Z′ are based on the use of test functions in class Z (the Fourier transforms of functions in D). Much, but not all, of chapter 12 can simply be adapted by replacing S by D (or Z) and S′ by D′ (or Z′). For instance, functionals of type D′ and Z′ are defined as follows.

Definition An association of exactly one real or complex number with each ϕ∈D is said to be a functional in class D′ if there exists at least one sequence of ordinary functions such that fnϕ∈L for each ϕ∈D and each n, and such that for each ϕ∈D the number associated with ϕ is equal to limn→∞ int; fnϕ.