Integrable Boehmians, Fourier Transforms, and Poisson’s Summation Formula

Boehmians are classes of generalized functions whose construction is algebraic. The first construction appeared in a paper that was published in 1981 [6]. In [8], P. Mikusiński constructs a space of Boehmians, βL1(R), in which each element has a Fourier transform. Mikusiński shows that the Fourier transform of a Boehmian satisfies some basic properties, and he also proves an inversion theorem. However, the range of the Fourier transform is not investigated. Also, Mikusiński states that βL1(R) contains some elements which are not Schwartz distributions, but no examples are given. We will address these problems in this paper. In this note, we will construct a space of Boehmians β`(R). The space of integrable functions on the real line can be identified with a proper subspace of β`(R). Each element of β`(R) has a Fourier transform which is a continuous function and satisfies a growth condition at infinity. Conditions are given which ensure that a given function is the Fourier transform of an element of β`(R).