Related papers: From "Dirac combs" to Fourier-positivity
Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a…
These brief lecture notes are intended mainly for undergraduate students in engineering or physics or mathematics who have met or will soon be meeting the Dirac delta function and some other objects related to it. These students might have…
We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls…
Properties of 2-dimensional generalizations of sine functions that are symmetric or antisymmetric with respect to permutation of their two variables are described. It is shown that the functions are orthogonal when integrated over a finite…
We investigate the problem of reconstructing a 2D piecewise smooth function from its bandlimited Fourier measurements. This is a well known and well studied problem with many real world implications, in particular in medical imaging. While…
We develop a general procedure for reduction along strong Dirac maps, which are a broad generalization of Poisson momentum maps. We recover a large number of familiar constructions in Poisson and quasi-Poisson geometry, and we introduce new…
This paper investigates the dimension theory of some families of continuous piecewise linear iterated function systems. For one family, we show that the Hausdorff dimension of the attractor is equal to the exponential growth rate obtained…
We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to ergodic theory, topological dynamics,…
It is shown that the cone in $\mathbb{R}^{d+1}$ has Fourier dimension $d-1$. This verifies a conjecture of Fraser and Kroon.
Given a Fourier transformable measure in two dimensions, we find a formula for the intensity of its Fourier transform along circles. In particular, we obtain a formula for the diffraction measure along a circle in terms of the…
A classical theorem of Fatou asserts that the Radon-Nikodym derivative of any finite positive Borel measure, $\mu$, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson…
In field theories one often works with the functionals which are integrals of some densities. These densities are defined up to divergence terms (boundary terms). A Poisson bracket of two functionals is also a functional, i.e., an integral…
We discuss the Dirac quantization of two dimensional gravity with bosonic matter fields. After defining the extended Hamiltonian it is possible to fix the gauge completely. The commutators can all be obtained in closed form; nevertheless,…
This notes explains how a standard algorithm that constructs the discrete Fourier transform has been formalised and proved correct in the Coq proof assistant using the SSReflect extension.
A Lagrange Theorem in dimension 2 is proved, for a particular two-dimensional algorithm, with a very natural geometrical definition. Dirichlet-type properties for the convergence of the algorithm are also proved. These properties procced…
Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and normal distributions for various functionals of the process.
We show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors. We then utilise this result to give a method for constructing the probability density…
Fourier series are considered on the one-dimensional torus for the space of periodic distributions that are the distributional derivative of a continuous function. This space of distributions is denoted $\alext$ and is a Banach space under…
We systematically find conditions which yield locally uniform convergence in the Fourier inversion formula in one and higher dimensions. We apply the gained knowledge to the complex inversion formula of the Laplace transform to extend known…
We study norm convergence and summability of Fourier series in the setting of reduced twisted group $C^*$-algebras of discrete groups. For amenable groups, F{\o}lner nets give the key to Fej\'er summation. We show that Abel-Poisson…