fractional lie series and transforms as canonical mappings dr. abd el-salam

Fractional, canonical, and simplified fractional cosine transforms

Fourier transform can be generalized into the fractional Fourier transform (FRFT), linear canonical transform (LCT), and simplified fractional Fourier transform (SFRFT). They extend the utilities of original Fourier transform, and can solve many problems that can’t be solved well by original Fourier transform. In this paper, we will generalize the cosine transform. We will derive fractional cos...

Mostafa Abd El Meguid

Optical implementations of two-dimensional fractional fourier transforms and linear canonical transforms with arbitrary parameters.

We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be con...

Poisson–Lie T–plurality as canonical transformation

We generalize the prescription realizing classical Poisson–Lie T–duality as canonical transformation to Poisson–Lie T–plurality. The key ingredient is the transformation of left–invariant fields under Poisson–Lie T–plurality. Explicit formulae realizing canonical transformation are presented and the preservation of canonical Poisson brackets and Hamiltonian density is shown. PACS: 02.30.Ik, 04....

Appell Transformation and Canonical Transforms

The interpretation of the optical Appell transformation, as previously elaborated in relation to the free-space paraxial propagation under both a rectangular and a circular cylindrical symmetry, is reviewed. Then, the caloric Appell transformation, well known in the theory of heat equation, is shown to be amenable for a similar interpretation involving the Laplace transform rather than the Four...

Fast linear canonical transforms.

The linear canonical transform provides a mathematical model of paraxial propagation though quadratic phase systems. We review the literature on numerical approximation of this transform, including discretization, sampling, and fast algorithms, and identify key results. We then propose a frequency-division fast linear canonical transform algorithm comparable to the Sande-Tukey fast Fourier tran...