Inversion of Umarov–Tsallis–Steinberg’s q-Fourier transform and the complex-plane generalization

We introduce a complex q -Fourier transform as a generalization of the (real) one analyzed in [S. Umarov, C. Tsallis, S. Steinberg, Milan J. Math. 307 (2008)]. By recourse to tempered ultradistributions we show that this complex-plane generalization overcomes all the troubles that afflict its real counterpart.