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68) where we leave out phases. Yet this transformation is only an approximation to the LCT, and it is generally not unitary. Regarding the spacing of the sampling points, Ding [67] has given a sampling theorem that generalizes that of Shannon in terms of the desired extent of the LCT transform signal. 67) is that Ž CM CM D 1; this occurs only for values of the parameter b such that 1=b is an integer relatively prime to N [68]. Combining both results, in [69], the authors present sufficient conditions on the sampling rate of fxm gNmD1 for any one LCT to ensure its unitarity.

56) Issues related to the meshing between Bargmann and LCT transforms were discussed in a think-tank at the Centre de Recherches Mathématiques (Université de Montréal) during the closing months of 1973. ItÁ was also noted that the real heat diffusion kernel to time t > 0 is the 10 12it complex LCTs with kernel p exp. , Á a b without inverses). ia=b/ < 0. If a is real, this means that the complex value of b must be in the lower complex half-plane, < arg b < 0. But unitarity is a cherished property among group theorists, so the question was posed to find appropriate Hilbert spaces to comply with this requirement.

Moshinsky, Linear canonical transformations and matrix elements. J. Math. Phys. 12, 1780–1783 (1971) 5. S. B. Wolf, Independent simultaneous discoveries visualized through network analysis: the case of Linear Canonical Transforms. Scientometrics (2015). 1007/s/11192-015-1602-x online 20/06 6. B. Wolf, Integral Transforms in Science and Engineering (Plenum Publishing Corporation, New York, 1979) 7. M. Nazarathy, J. Shamir, Fourier optics described by operator algebra. J. Opt. Soc. Am. 70, 150–159 (1980) 8.

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