Recent developments in the theory of the fractional Fourier and linear canonical transforms

In recent years, there has been an enormous effort put in the definition and analysis of fractional or fractal operators. Fractional calculus is for example a flourishing field of active research. In this paper we restrict ourselves to the fractional Fourier operator and friends that are traditionally used in optics, mechanical engineering and signal processing. The book by H.M. Ozaktas, Z. Zalevsky, and M.A. Kutay, The, fractional Fourier transform, John Wiley, 2001 gives a state of the art of 2001. Because this field is still in full expansion, we want to summarize in this survey paper some of the recent developments that appeared in the literature since then, revealing some unexplored aspects.