Related papers: Fourier transforms on Cantor sets: A study in non-…
We present a general theory of fractal transformations and show how it leads to a new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves…
Recently, there has been an increasing interest in the study of hypercomplex signals and their Fourier transforms. This paper aims to study such integral transforms from general principles, using 4 different yet equivalent definitions of…
The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized…
This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier…
In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems…
In this paper, we give a review of fractal calculus which is an expansion of standard calculus. Fractal calculus is applied for functions which are not differentiable or integrable on totally disconnected fractal sets such as middle-$\mu$…
We consider an optical diffraction grating in which the spatial distribution of open slits forms a fractal set. The Fraunhofer diffraction patterns through the fractal grating are obtained analytically for the simplest triad Cantor type and…
In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under…
In this contribution we generalize the classical Fourier Mellin transform [S. Dorrode and F. Ghorbel, Robust and efficient Fourier-Mellin transform approximations for gray-level image reconstruction and complete invariant description,…
As a generalization of the Fourier transform, the fractional Fourier transform was introduced and has been further investigated both in theory and in applications of signal processing. We obtain a sampling theorem on shift-invariant spaces…
This report will explore and answer fundamental questions about taking Fourier Transforms and tying it with recent advances in AI and neural architecture. One interpretation of the Fourier Transform is decomposing a signal into its…
We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate…
The quaternion Fourier transform (QFT), a generalization of the classical 2D Fourier transform, plays an increasingly active role in particular signal and colour image processing. There tends to be an inordinate degree of interest placed on…
An experiment to demonstrate the Fourier transform of an electric signal using the Kundt's tube is described. The results of finding the component frequencies and an approximation to the amplitudes of two sinusoidal signals which compose an…
The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…
We introduce the notion of boundary representation for fractal Fourier expansions, starting with a familiar notion of spectral pairs for affine fractal measures. Specializing to one dimension, we establish boundary representations for these…
We propose a discrete fractional random transform based on a generalization of the discrete fractional Fourier transform with an intrinsic randomness. Such discrete fractional random transform inheres excellent mathematical properties of…
The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT…
We establish properties of a new type of fractal which has partial self similarity at all scales. For any collection of iterated functions systems with an associated probability distribution and any positive integer V there is a…
A set of semi-analytical techniques based on Fourier analysis is used to solve wave scattering problems in variously shaped waveguides with varying normal admittance boundary conditions. Key components are newly developed conformal mapping…