On the Fourier transform and the exchange property

Since Boehmians were introduced, extensions of the Fourier transform to spaces of Boehmians attracted a lot of attention (see [2, 3, 4, 5, 6, 7, 8, 9]). In some cases, the range of the extended Fourier transform is a space of functions. In other constructions, the range is a space of distributions or a space of Boehmians. In this paper, we would like to consider the space of tempered Boehmians presented in [8]. In this case, the range of the Fourier transform is the space of all distributions ′. This work is motivated by [1]. First we recall briefly the construction of the space of tempered Boehmians. A continuous function f : RN → C is called slowly increasing if there is a polynomial p on RN such that | f (x)| ≤ p(x) for all x ∈ RN . The space of slowly increasing functions will be denoted by (RN ) or simply . An infinitely differentiable function f :RN → C is called rapidly decreasing if