Ahmed I. Zayed

Advances in Shannon's sampling theory

Introduction and a Historical Overview. Shannon Sampling Theorem and Band-Limited Signals. Generalizations of Shannon Sampling Theorems. Sampling Theorems Associated with Sturm-Liouville Boundary-Value Problems. Sampling Theorems Associated with Self-Adjoint Boundary-Value Problems. Sampling by Using Greens Function. Sampling Theorems and Special Functions. Kramers Sampling Theorem and Lagrange-Type Interpolation in N Dimensions. Sampling Theorems for Multidimensional Signals-The Feichtinger-Grochenig Sampling Theory. Frames and Wavelets: A New Perspective on Sampling Theorems.

A convolution and product theorem for the fractional Fourier transform

The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.

Hilbert transform associated with the fractional Fourier transform

The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t). We generalize the definition of the Hilbert transform in order to obtain the analytic part of a signal that is associated with its fractional Fourier transform, i.e., that part of the signal f(t) that is obtained by suppressing the negative frequency content of the fractional Fourier transform of f(t). We also show that the generalized Hilbert transform has similar properties to those of the ordinary Hilbert transform, but it lacks the semigroup property of the fractional Fourier transform.

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

mmThere are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems

This article is devoted to a connection between Kramer’s sampling theorem and sampling expansions generated by Lagrange interpolation. It is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm–Liouville boundary-value problem. This new approach, which for a variety of regular and singular Sturm–Liouville problems leads to associated sampling theorems, recovers not only many known sampling expansions but also gives new ways to calculate the corresponding sampling functions. New sampling series are included.

On Kramer's sampling theorem associated with general Strum-Liouville problems and Lagrange interpolation

Kramer’s sampling theorem (which is a generalization of Shannon’s sampling theorem) and its relationship with Lagrange interpolation is studied. Kramer’s sampling theorem is extended to the case where the interval

New sampling formulae for the fractional Fourier transform

( {a,b} )

Sampling, Wavelets, and Tomography

is infinite and it is shown that any function that has a sampling expansion in the scope of Kramer’s theorem also has a Lagrange-type interpolation expansion, provided that the kernel associated with Kramer’s theorem arises from a second-order Sturm-Liouville boundary value problem, whether it is regular or singular. This new technique not only reproduces many of the known sampling expansions such as those for the cosine, the finite Hankel, and the continuous Legendre transforms, but also generates new ones such as those for the continuous Jacobi, Laguerre, and Hermite transforms.

A generalization of the prolate spheroidal wave functions

In this note we obtain two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense. In both cases, we use samples from both the signal and its Hilbert transform, but each taken at half the Nyquist rate.

Shift-Invariant and Sampling Spaces Associated With the Fractional Fourier Transform Domain

Sampling, wavelets, and tomography are three active areas of contemporary mathematics sharing common roots that lie at the heart of harmonic and Fourier analysis. The advent of new techniques in mathematical analysis has strengthened their interdependence and led to some new and interesting results in the field. This state-of-the-art book not only presents new results in these research areas, but it also demonstrates the role of sampling in both wavelet theory and tomography. Specific topics covered include: * Robustness of Regular Sampling in Sobolev Algebras * Irregular and Semi-Irregular Weyl-Heisenberg Frames * Adaptive Irregular Sampling in Meshfree Flow Simulation * Sampling Theorems for Non-Bandlimited Signals * Polynomial Matrix Factorization, Multidimensional Filter Banks, and Wavelets * Generalized Frame Multiresolution Analysis of Abstract Hilbert Spaces * Sampling Theory and Parallel-Beam Tomography * Thin-Plate Spline Interpolation in Medical Imaging * Filtered Back-Projection Algorithms for Spiral Cone Computed Tomography Aimed at mathematicians, scientists, and engineers working in signal and image processing and medical imaging, the work is designed to be accessible to an audience with diverse mathematical backgrounds. Although the volume reflects the contributions of renowned mathematicians and engineers, each chapter has an expository introduction written for the non-specialist. One of the key features of the book is an introductory chapter stressing the interdependence of the three main areas covered. A comprehensive index completes the work. Contributors: J.J. Benedetto, N.K. Bose, P.G. Casazza, Y.C. Eldar, H.G. Feichtinger, A. Faridani, A. Iske, S. Jaffard, A. Katsevich, S. Lertrattanapanich, G. Lauritsch, B. Mair, M. Papadakis, P.P. Vaidyanathan, T. Werther, D.C. Wilson, A.I. Zayed