COLLOQUIUM MATHEMATICUM VOL. 108 2007 NO. 2 ON THE FOURIER TRANSFORM, BOEHMIANS, AND DISTRIBUTIONS BY DRAGU ATANASIU (Bor˚ as) and PIOTR MIKUSI ´ NSKI (Orlando, FL) Abstract. We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform. 1. Introduction. The Fourier transform is one of the most important tools in analysis. When using the Fourier transform on a space of functions or generalized functions, knowing the range is usually essential. For this reason, the space of square integrable functions and the space of tempered distributions are very useful. However, the space of square integrable func- tions is often too small, while the space of tempered distributions requires substantial machinery from functional analysis. In this paper we consider a number of spaces of generalized functions for which the Fourier transform can be defined in a simple manner and the range can be easily characterized. The constructions have algebraic char- acter, which means that the definitions do not require topological consid- erations. On the other hand, the spaces have natural topologies that have desirable properties. Some of the relevant objects are examples of the so-called “generalized quotients” (see [6] and [3]), other are examples of Boehmians. One of the spaces of Boehmians is isomorphic to the space of all Schwartz distribu- tions. The space B(L 2 (R N ), G ) is an example of the so-called generalized quo- tients. We start by recalling essential details of the construction of general- ized quotients. Let X be a nonempty set and let G be a commutative semigroup acting on X injectively. This means that every ϕ ∈ G is an injective map ϕ : X → X and (ϕψ)x = ϕ(ψx) for all ϕ,ψ ∈ G and x ∈ X . 2000 Mathematics Subject Classification : Primary 42B10, 44A40; Secondary 44A35, 46F12. Key words and phrases : convolution quotients, Fourier transform. [263] c  Instytut Matematyczny PAN, 2007