Let \(F_{(h,\nu)}\) be the space of differentiable functions \(\varphi(x)\) for which some sup-norm is finite. Then one has the continuous embedding \(F_{(h,\nu)}\subset F_{(h',\nu')}\), \(h\geq h'>0, \nu\geq\nu'\). Let \(\mathcal G=\bigcap_{h,\nu}F_{(h,\nu)}\) be endowed with the natural projective topology. Then \(\mathcal G\) is a Fréchet space and its dual \(\mathcal G'\) is called the space of extended Fourier hyperfunctions. The author shows that \(\mathcal G\) can be embedded as a weakly dense subspace of its dual. It is shown that \(\mathcal G \ast \mathcal G ' \subset \mathcal G\) and that the convolution of two test functions with an extended Fourier hyperfunction is associative. The Fourier transform is also studied.