A unified group theoretical method for the partial Fourier analysis on semi-direct product of locally compact groups (Q2258335)

The author defines the \(\tau\)-dual group (partial dual group) \(G_{ \hat{\tau}}\) of \(G_{\tau }=H\ltimes _{\tau }K\). This group is the semi-direct product of \(H\) and \(\hat{K}\) with respect to the continuous homomorphism \(\hat{\tau}:H\rightarrow \mathrm{Aut}\left( \hat{K}\right) \) and it is given by \(\hat{\tau}_{h}\left( \omega \right) :=\omega \circ \tau _{h^{-1}}\) for all \( h\in H\) and \(\omega \in \hat{K}\). Then the author generalizes the Pontrjagin duality theorem which says that the \(\hat{\tau}\)-dual group and \(G_{\tau }\) are isomorphic (see the third section). In the fourth section, the author defines the \(\tau\)-Fourier transform. Then for this transform, a Parseval formula is proved. So it is shown that the \(\tau\)-Fourier transform is a unitary transform from \(L^{2}\left( G_{\tau }\right) \) onto \(L^{2}\left( G_{\tau }\right) \). Finally, the author proves an inversion formula for the \( \tau\)-Fourier transform. In the last section, the author considers the affine group \(ax+b\), Euclidean groups and the Weyl-Heisenberg group for the \(\tau\)-Fourier transform. In this paper, there is a unified group theoretical approach to partial Fourier analysis on semi-direct products of locally compact groups. Since the partial Fourier analysis has many applications in science and engineering problems this paper is highly important.