An optical implementation of the fractional Fourier transform (FRT) with broadband illumination is proposed by use of a single imaging element, namely, a blazed diffractive lens. The setup displays an achromatized version of the FRT of order P of any two-dimensional input function. This fractional order can be tuned continuously by shifting of the input along the optical axis. Our compact and f...

The 2-D Fourier transform has been generalized into the 2-D separable fractional Fourier transform (replaces 1-D Fourier transform by 1-D fractional Fourier transform for each variable) and the 2-D separable canonical transform (further replaces the fractional Fourier transform by canonical transform) in [3]. It also has been generalized into the 2-D unseparable fractional Fourier transform wit...

In this paper, the combination of conduction with radiation into a semitransparent medium which includes absorption, emission and scattering has been investigated. In order to Study the conduction in medium, the Non-Fourier heat conduction has been applied. In this model there is a time delay between heat flux and temperature gradient. Also, in contrast with Fourier heat conduction, the speed o...

In the next two lectures, we will continue to expand our toolbox for computing in frequency domain, by adding two important computational tools. The first, the Fast Fourier Transform (FFT) is a family of algorithms for efficiently computing the Discrete Fourier Transform. These algorithms are of great practical importance, as theymake it possible to perform frequency domain analysis of very lar...

AMS Subject Classi cation: 20-04, 05C25, 20C30 Rockmore was supported in part by a National Science Foundation Mathematical Sciences Postdoctoral Fellowship. We study the complexity of performing Fourier analysis for the group SL2(Fq), where Fq is the finite field of q elements. Direct computation of a complete set of Fourier transforms for a complex-valued function f on SL2(Fq) requires q6 ope...

It is shown that number theoretic transforms (NTT) can be used to compute discrete Fourier transform (DFT) very efficiently. By noting some simple properties of number theory and the DFT, the total number of real multiplications for a length-P DFT is reduced to (P — 1). This requires less than one real multiplication per point. For a proper choice of transform length and NTT, the number of shif...