Related papers: On the diagonalization of the discrete Fourier tra…
Using the shift-operator technique, a compact formula for the Fourier transform of a product of two Slater-type orbitals located on different atomic centers is derived. The result is valid for arbitrary quantum numbers and was found to be…
We analyze the Double Fourier Sphere (DFS) method on the rotation group $\mathcal{SO}(3)$ in the frequency domain and demonstrate its central role in fast algorithms. Fast Fourier algorithms on $\mathcal{SO}(3)$ are commonly formulated as a…
A Fourier transform S is defined for the quantum double D(G) of a finite group G. Acting on characters of D(G), S and the central ribbon element of D(G) generate a unitary matrix representation of the group SL(2,Z). The characters form a…
Two algorithms are introduced for the computation of discrete integral transforms with a multiscale approach operating in discrete three-dimensional (3D) volumes while considering its real-time implementation. The first algorithm, referred…
The Fourier transform is approximated over a finite domain using a Riemann sum. This Riemann sum is then expressed in terms of the discrete Fourier transform, which allows the sum to be computed with a fast Fourier transform algorithm more…
The FFT algorithm that implements the discrete Fourier transform is considered one of the top ten algorithms of the $20$th century. Its main strengths are the low computational cost of $\mathcal{O}(n \log n$) and its stability. It is one of…
Edge devices are being deployed at increasing volumes to sense and act on information from the physical world. The discrete Fourier transform (DFT) is often necessary to make this sensed data suitable for further processing -- such as by…
A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from $\ZZ[i]$ to $\ZZ[i]$, whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be…
The hopping matrix element $t$ quantifies the quantum-mechanical coupling between frontier orbitals on a pair of nearby molecules. Convenient and generally applicable methods to determine $t$ from DFT calculations are lacking; $t$ can be…
The composition of the Fourier transform in $\mathbb{R}^n$ with a suitable pseudodifferential operator is called a Fourier operator. It is compact in appropriate function spaces. The paper deals with its spectral theory. This is based on…
Fourier representation (FR) is an indispensable mathematical formulation for modeling and analysis of physical phenomenon, engineering systems and signals in numerous applications. In this study, we present the generalized Fourier…
Time reversal and spin flip are discrete symmetry operations of substantial import to quantum information and quantum computation. Spin flip arises in the context of separability, quantification of entanglement and the construction of…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
We introduce an algebra $\mathcal H$ consisting of difference-reflection operators and multiplication operators that can be considered as a $q=1$ analogue of Sahi's double affine Hecke algebra related to the affine root system of type…
The fast Fourier transform, FFT, is a useful and prevalent algorithm in signal processing. It characterizes the spectral components of a signal, or is used in combination with other operations to perform more complex computations such as…
In this paper we show that the discrete Fourier transform can be performed by scattering a coherent particle or laser beam off a two-dimensional potential that has the shape of rings or peaks. After encoding the initial vector into the…
The Z Transform is a mathematical operation in signal processing, which gives a tractable way to solve linear, constant-coefficient difference equations. Based on the classical Z transform and inspired by the thought of sliding DFT, a new…
Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov-Shabat spectral problem, this article delivers new…
The nonlinear Fourier transform (NLFT) extends the classical Fourier transform by replacing addition with matrix multiplication. While the NLFT on $\mathrm{SU}(1,1)$ has been widely studied, its $\mathrm{SU}(2)$ variant has only recently…
We introduce the notion of discrete Baker-Akhiezer (DBA) modules, which are modules over the ring of difference operators, as a certain discretization of Baker-Akhiezer modules which are modules over the ring of differential operators. We…