ON THE FOURIER TRANSFORM AND THE EXCHANGE PROPERTY DRAGU ATANASIU AND PIOTR MIKUSI ´ NSKI Received 19 June 2005 A simplified construction of tempered Boehmians is presented. The new construction shows that considering delta sequences and convergence arguments is not essential. 1. Introduction 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 con- tinuous function f : R N → C is called slowly increasing if there is a polynomial p on R N such that | f (x)|≤ p(x) for all x ∈ R N . The space of slowly increasing functions will be denoted by (R N ) or simply . An infinitely differentiable function f : R N → C is called rapidly decreasing if sup |α|≤m sup x∈R N ( 1+ x 2 1 + ··· + x 2 N ) m   D α f (x)   < ∞ (1.1) for every nonnegative integer m, where x = (x 1 , ... , x N ), α = (α 1 , ... , α N ), α n ’s are nonneg- ative integers, |α|= α 1 + ··· + α N , and D α = ∂ |α| ∂x α = ∂ |α| ∂x α1 1 ··· ∂x αN N . (1.2) The space of rapidly decreasing functions is denoted by (R N ) or simply . If f ∈  and ϕ ∈ , then the convolution f ∗ ϕ(x) =  R N f ( y)ϕ(x − y)dy (1.3) is well defined and f ∗ ϕ ∈ . Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:16 (2005) 2579–2584 DOI: 10.1155/IJMMS.2005.2579