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Discrete analogs of the index transforms, involving Bessel and the modified Bessel functions are introduced and investigated. The corresponding inversion theorems for suitable classes of functions and sequences are established.
We perform Bell's measurement and perform quantum Fourier transform with the classical vortex beam. The violation of Bell's inequality for such a non-separable classical correlation has been demonstrated experimentally. Based on the…
Using Bauer's expansion and properties of spherical Bessel and Legender functions, we deduce a new transform and briefly indicate its use.
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structure of functions, especially periodicity. The fact that Fourier transforms can also be used to…
The achievable data rates of current fiber-optic wavelength-division-multiplexing (WDM) systems are limited by nonlinear interactions between different subchannels. Recently, it was thus proposed to replace the conventional Fourier…
Let $f$ be a function on the real line. The Fourier transform inversion theorem is proved under the assumption that $f$ is absolutely continuous such that $f$ and $f'$ are Lebesgue integrable. A function $g$ is defined by…
We present a fast algorithm for generating full sky, high resolution ($\sim 5'$) simulations of the CMB anisotropy pattern. We also discuss the inverse problem, that of evaluating from such a map the full set of $a_{\ell m}$'s and the…
Bayesian methods which utilize Bayes' theorem to update the knowledge of desired parameters after each measurement, are used in a wide range of quantum science. For various applications in quantum science, efficiently and accurately…
This paper describes the Direct Fourier Permuation Algorithm, an efficient method of computing Bit Reversal of natural indices [1, 2, 3, ..., 2^k] in a vectorial manner (k iterations) and also proposes the Vectorial Digit Reversal…
We introduce a fast Fourier transform on regular d-dimensional lattices. We investigate properties of congruence class representants, i.e. their ordering, to classify directions and derive a Cooley-Tukey-Algorithm. Despite the fast Fourier…
The paper is concerned with efficient numerical methods for solving a linear system $\phi(A) x= b$, where $\phi(z)$ is a $\phi$-function and $A\in \mathbb R^{N\times N}$. In particular in this work we are interested in the computation of…
We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is…
We develop an exact wavelet transform on the three-dimensional ball (i.e. on the solid sphere), which we name the flaglet transform. For this purpose we first construct an exact transform on the radial half-line using damped Laguerre…
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which…
In this paper we derive and analyze an algorithm for inverting quaternion matrices. The algorithm is an analogue of the Frobenius algorithm for the complex matrix inversion. On the theory side, we prove that our algorithm is more efficient…
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…
Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups…
In this work the numerical solution of acoustic tomography problem based on the iterative and functional-analytical algorithms is considered. The mathematical properties of these algorithms were previously described in works of R.G.Novikov…
The proof of the theorem concerning to the inverse cyclotomic Discrete Fourier Transform algorithm over finite field is provided.
Two methods for fast Fourier transforms are used in a quantum context. The first method is for systems with dimension of the Hilbert space $D=d^n$ with $d$ an odd integer, and is inspired by the Cooley-Tukey formalism. The `large Fourier…