Fast computation of two-dimensional (2-D) generalized discrete Fourier transforms (GDFTs) and generalized discrete Hartley transforms (GDHTs) are discussed in terms of a new method which is derived based on index permutation, linear congruences, and polynomial transforms. Further, the 2-D skew circular convolution computed by the proposed skew polynomial transform is also included. ( 2000 Elsev...

In this paper we present new generalized convolutions with weight-function associated with the Fourier and Hartley transforms, and consider applications. Namely, using the generalized convolutions, we construct normed rings on the space L(R), provide the sufficient and necessary condition for the solvability of a class of integral equations of convolution type, and receive the explicit solution...

The general goal of this paper is to extend the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections without rebinning the divergent fan-beam and cone-beam projections into parallel-beam projections directly. The basic idea is to establish a novel link between the local Fourier transform of the projection data and the Fourier transform of the image object. Anal...

The theorem of sampling formulae has been deduced for band-limited or time-limited signals in the fractional Fourier domain by different authors. Even though the properties and applications of these formulae have been studied extensively in the literature, none of the research papers throw light on the Poisson sum formula and non-band-limited signals associated with the fractional Fourier trans...

‎By observing the equivalence of assertions on determining the jump of a‎ ‎function by its differentiated or integrated Fourier series‎, ‎we generalize a‎ ‎previous result of Kvernadze‎, ‎Hagstrom and Shapiro to the whole class of‎ ‎functions of harmonic bounded variation‎. ‎This is achieved without the finiteness assumption on‎ ‎the number of discontinuities‎. ‎Two results on determination of ...

Our aim in this paper is to prove an analog of Younis's Theorem on the image under the Jacobi transform of a class functions satisfying a generalized Dini-Lipschitz condition in the space $mathrm{L}_{(alpha,beta)}^{p}(mathbb{R}^{+})$, $(1< pleq 2)$. It is a version of Titchmarsh's theorem on the description of the image under the Fourier transform of a class of functions satisfying the Dini-Lip...

This brief explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a band-limited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem whi...

The propagation of mutual intensity through quadratic graded-index media or free space can be expressed in terms of two-dimensional fractional Fourier transforms for one-dimensional systems and in terms of fourdimensional fractional Fourier transforms for two-dimensional systems. As light propagates, its mutual intensity distribution is continually fractional Fourier transformed. These results ...

This brief explores the connections between nonuniform sampling of a certain function and the almost periodic extension of its Fourier transform. It is shown that the Fourier transform of a bandlimited function can be extended (as a weighted sum of translates) as a Stepanoff almost periodic function, to the whole frequency axis. This result leads to a generalized nonuniform sampling theorem whi...