Appell Transformation and Canonical Transforms

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...

Exponential Transforms and Appell Polynomials.

Appell Sequences, Continuous Wavelet Transforms and Series Expansions

A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives...

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...

Fourier-Mukai transforms and canonical divisors

Let X be a smooth projective variety. We study a relationship between the derived category of X and that of a canonical divisor. As an application, we will study FourierMukai transforms when κ(X) = dimX − 1.