You can define the Fourier transform of any hyperfunction and thus of any distribution: this was proved by Alexander Georgievich Smirnov in 2004 [1]. The details of the construction are quite technical so, in order to have an intuitive understanding of the methods he used, in my opinion the best thing to do is to read the abstract of [1], reported below
A new generalized function space in which all Gelfand–Shilov classes $S^{\prime0}_\alpha$ ($\alpha> 1$) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on $\Bbb R^k$ is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley–Wiener–Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformations is established.
Also the Zbl review of [1] is useful in order to understand the Author's construction.
Reference
[1] Alexander Georgievich Smirnov, "Fourier transformation of Sato's hyperfunctions", Advances in Mathematics, Vol. 196, No. 2, pp. 310-345 (2005), DOI:10.1016/j.aim.2004.09.003, MR2166310, Zbl 1087.46031.
