In this paper we describe a method for computing the Discrete Fourier Transform (DFT) of a sequence of n elements over a finite field GF(pm) with a number of bit operations 0(nm log(nm) ■ P(q)) where P(q) is the number of bit operations required to multiply two q-bit integers and q = 2 log2« + 4 log2m + 4 log2p. This method is uniformly applicable to all instances and its order of complexity is...

In this paper, we propose an optical image watermarking scheme using the fractional Fourier transform. A two-dimensional watermark is encrypted using a double random fractional order Fourier domain encoding technique. The encrypted image is watermarked into a three-dimensional intensity image reconstructed from a real in-line digital hologram. The watermark is recovered by applying the correspo...

We review the Fourier-Laguerre transform, an alternative harmonic analysis on the three-dimensional ball to the usual Fourier-Bessel transform. The Fourier-Laguerre transform exhibits an exact quadrature rule and thus leads to a sampling theorem on the ball. We study the definition of convolution on the ball in this context, showing explicitly how translation on the radial line may be viewed as...

A straightforward discretisation of high-dimensional problems often leads to a curse of dimensions and thus the use of sparsity has become a popular tool. Efficient algorithms like the fast Fourier transform (FFT) have to be customised to these thinner discretisations and we focus on two major topics regarding the Fourier analysis of high-dimensional functions: We present stable and effective a...

We show theoretically and experimentally that two-photon coherent control in a V -shape three-level system projects a one-photon coherent transient in a simple two-level system. The secondand third-order spectral phase terms of a shaped laser pulse play the roles of time and quadratic spectral phase, respectively, in conventional coherent transients. In a three-pulse coherent control experiment...

This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation requires N = 2 p 3 q 5 r. The communication required is the same as for a transform of length N = ...

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions o...