Schwartz Distributions Research Articles

Solving wave equations in the space of Schwartz distributions: the beauty of generalised functions in physics

Solving wave equations in the space of Schwartz distributions: the beauty of generalised functions in physics

On the convergence of Fourier representations and Schwartz distributions

While Fourier theory remains a cornerstone for analyzing and interpreting the spectral content of diverse signals, its limitations often surface in practical applications. Notably, popular signals like sinusoids, Dirac deltas, signums, and unit steps lack convergent Fourier representations within the conventional framework. This discrepancy necessitates distribution theory to build and interpret suitable representations for such signals. However, existing literature in signal processing and communication engineering often glosses over these intricacies, leaving researchers grappling with obscure concepts regarding the very existence of Fourier representations for non-conforming signals. This work bridges this critical gap by offering a comprehensive exploration of the conditions guaranteeing the existence of Fourier representations. We introduce a novel linear space – the Gauss–Schwartz (GS) function space – and its corresponding class of tempered superexponential (TSE) distributions. We demonstrate that the Fourier transform (FT) acts as an isomorphism on the GS space of test functions and, by duality, on TSE distributions. Crucially, the GS space proves to be minimal in the sense that its dual, encompassing TSE distributions, represents the largest possible linear space over which the FT can be defined through duality. This theoretical advancement signifies a twofold contribution. Firstly, it clarifies the existence and interpretation of Fourier representations for prevalent signals that evade conventional analysis. Secondly, the introduction of the GS-TSE framework expands the reach of the FT to encompass a broader spectrum of functions, enabling novel applications in diverse fields. Ultimately, this work paves the way for a more robust and accessible understanding of Fourier analysis, empowering researchers and practitioners to leverage its full potential in exploring and processing a wider range of signals.

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

Products of distributions in Colombeau algebra

In this paper, we evaluate some products of distributions in Colombeau algebra of generalized functions. In the classical theory of Schwartz distributions, multiplication of distributions is not defined for two arbitrary singular distributions. The properties of the Colombeau algebra allow us to calculate products of singular distributions which are not defined in the classical theory. The notion of association in Colombeau algebra of generalized functions allows us the results obtained in this way to be considered as products in the classical theory of distributions. The definition of the Colombeau product of distributions can be considered as generalization of their classical product in Schwartz theory.

Moment tracking and their coordinate transformations for macroparticles with an application to plasmas around black holes

Particle-in-cell (PIC) codes usually represent large groups of particles as a single macroparticle. These codes are computationally efficient but lose information about the internal structure of the macroparticle. To improve the accuracy of these codes, this work presents a method in which, as well as tracking the macroparticle, the moments of the macroparticle are also tracked. Although the equations needed to track these moments are known, the coordinate transformations for moments where the space and time coordinates are mixed cannot be calculated using the standard method for representing moments. These coordinate transformations are important in astrophysical plasma, where there is no preferred coordinate system. This work uses the language of Schwartz distributions to calculate the coordinate transformations of moments. Both the moment tracking and coordinate transformation equations are tested by modelling the motion of uncharged particles in a circular orbit around a black hole in both Schwarzschild and Kruskal–Szekeres coordinates. Numerical testing shows that the error in tracking moments is small, and scales quadratically. This error can be improved by including higher order moments. By choosing an appropriate method for using these moments to deposit the charge back onto the grid, a full PIC code can be developed.

Vibration modes of the Euler–Bernoulli beam equation with singularities

We consider the time dependent Euler–Bernoulli beam equation with discontinuous and singular coefficients. Using an extension of the Hörmander product of distributions with non-intersecting singular supports (L. Hörmander, 1983 [25]), we obtain an explicit formulation of the differential problem which is strictly defined within the space of Schwartz distributions. We determine the general structure of its separable solutions and prove existence, uniqueness and regularity results under quite general conditions. This formalism is used to study the dynamics of an Euler–Bernoulli beam model with discontinuous flexural stiffness and structural cracks. We consider the cases of simply supported and clamped-clamped boundary conditions and study the relation between the characteristic frequencies of the beam and the position, magnitude and structure of the singularities in the flexural stiffness. Our results are compared with some recent formulations of the same problem.

An existence and uniqueness result about algebras of Schwartz distributions

We prove that there exists essentially one minimal differential algebra of distributions A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}, satisfying all the properties stated in the Schwartz impossibility result [L. Schwartz, Sur l’impossibilité de la multiplication des distributions, 1954], and such that Cp∞⊆A⊆D′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C_p^{\\infty } \\subseteq \\mathcal A\\subseteq \\mathcal D' $$\\end{document} (where Cp∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal C_p^{\\infty }$$\\end{document} is the set of piecewise smooth functions and D′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal D'$$\\end{document} is the set of Schwartz distributions over R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb R$$\\end{document}). This algebra is endowed with a multiplicative product of distributions, which is a generalization of the product defined in [N.C.Dias, J.N.Prata, A multiplicative product of distributions and a class of ordinary differential equations with distributional coefficients, 2009]. If the algebra is not minimal, but satisfies the previous conditions, is closed under anti-differentiation and the dual product by smooth functions, and the distributional product is continuous at zero then it is necessarily an extension of A\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal A$$\\end{document}.

Relativistic Free Schrödinger Equation for Massive Particles in Schwartz Distribution Spaces

In this work, we pose and solve, in tempered distribution spaces, an open problem proposed by Schrödinger in 1925. In particular, on the Schwartz distribution spaces, we define the linear continuous quantum operators associated with relativistic Hamiltonians of massive particles—particles with rest mass different from 0 and evolving in the four-dimensional Minkowski vector space M4. In other words, upon the tempered distribution state-space S′(M4,C), we have found the most natural way to introduce the free-particle relativistic Hamiltonian operator and its corresponding Schrödinger equation (together with its conjugate equation, standing for antiparticles). We have found the entire solution space of our relativistic linear continuous evolution equation by completely solving a division problem in tempered distribution space. We define the Hamiltonian (Schwartz diagonalizable) operator as the principal square root of a strictly positive, Schwartz diagonalizable second-order differential operator (linked with the “Klein–Gordon operator” on the tempered distribution space S4′). The principal square root of a Schwartz nondefective operator is defined in a straightforward way—following the heuristic fashion of some classic and greatly efficient quantum theoretical approach—in the paper itself.

Asymptotic Almost Automorphy for Algebras of Generalized Functions

The paper aims to to study the concept of asymptotic almost automorphy in the context of generalized functions. We introduce an algebra of asymptotically almost automorphic generalized functions which contains the space of smooth asymptotically almost automorphic functions as a subalgebra. The fundamental importance of this algebra, is related to the impossibility of multiplication of distributions; it also contains the asymptotically almost automorphic Sobolev--Schwartz distributions as a subspace. Moreover, it is shown that the introduced algebra is stable under some nonlinear operations. As a by pass result, the paper gives a Seeley type result on extension of functions in the context of the algebra of bounded generalized functions and the algebra of bounded generalized functions vanishing at infinity, these results are used to prove the fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. As applications, neutral difference-differential systems are considered in the framework of the algebra of generalized functions.

Sato hyperfunctions via relative Dolbeault cohomology

The relative Dolbeault cohomology which naturally comes up in the theory of Čech–Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of Grothendieck and Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

Coherent Schwartz distributions of the Heisenberg algebra and inverted oscillator

An unusual family of generalized coherent states of the Heisenberg algebra is constructed. It consists of two families of functionals on the Schwartz space. Each of them has the key properties of ordinary coherent states: it intertwines the representations of a given algebra, has the completeness property, and minimizes the product of variances in the Heisenberg relation. However, unlike the usual coherent states that are eigenstates for the annihilation operator, the constructed distributions belong to the continuous spectrum of Hermitian generators of the Heisenberg algebra. The inner product of functionals from different families has the main properties of the overlap function of ordinary coherent states: it is continuous in parameters, satisfies the reproducing identity, and has the corresponding geometric meaning. The application of unusual coherent states of the continuous spectrum is demonstrated by an example of solving the spectral problem for an inverted oscillator.

Besov regularity in non-linear generalized functions

We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise criteria for detecting Besov regularity of distributions.

A numerical scheme for stochastic differential equations with distributional drift

In this paper we introduce a scheme for the numerical solution of one-dimensional stochastic differential equations (SDEs) whose drift belongs to a fractional Sobolev space of negative regularity (a subspace of Schwartz distributions). We obtain a convergence rate in a suitable L1-norm and, as a by-product, a convergence rate for a numerical scheme applied to SDEs with drift in Lp-spaces with p∈(1,∞).

Global solvability and propagation of regularity of sums of squares on compact manifolds

We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds T × G, where G is further assumed to be a Lie group. As in a recent article due to the authors, our analysis is carried out in terms of a system of left-invariant vector fields on G naturally associated with the operator under study, a simpler object which nevertheless conveys enough information about the original operator so as to fully encode its solvability. As a welcome side effect of the tools developed for our main purpose, we easily prove a general result on propagation of regularity for such operators.

Recursive integral equations for random weights averages: Exponential functions and Cauchy distribution

In this article, firstly, we prove that functions of the form ϕ(x)=ecxI(−∞,0)(x)+ebxI[0,+∞)(x), c,b constants, are the only solutions to the integral equation ϕ(x)=∫01ϕ(ux)ϕ((1−u)x)du. This indeed gives the result of Van Asshe (1987) who used the Schwartz distribution theory to prove that for i.i.d X and Y, UX+(1−U)Y=dX if and only if X has a Cauchy distribution. Secondly, by looking into certain recursive integral equations involving characteristic functions, we prove that if for an n≥2, the random weight mean U(1)X1+(U(2)−U(1))X2+⋯+(1−U(n−1))Xn has a Cauchy distribution, then X1 has a Cauchy distribution; random variables X1,…,Xn are i.i.d, the random weights are the cuts of (0,1) by a uniform sample. The multivariate analogue of this result is also provided.

A new algebra of distributions; initial-value problems involving Schwartz distributions. I.

A new algebra of distributions; initial-value problems involving Schwartz distributions. I.

A new algebra of distributions; intial value problems involving Schwartz distributions. II.

A new algebra of distributions; intial value problems involving Schwartz distributions. II.

Continuous wavelet transform of Schwartz distributions in 𝒟′(ℝ𝑛), 𝑛 ≤ 1

Abstract In this paper, we extend the continuous wavelet transform to Schwartz distributions in D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , and derive the corresponding wavelet inversion formula (valid modulo a constant distribution) interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is an element ψ ⁢ ( x ) \psi(x) of D ⁢ ( R n ) \mathcal{D}(\mathbb{R}^{n}) , n ≥ 1 n\geq 1 , which, when integrated along each of the real axes X 1 , X 2 , X 3 , … , X n X_{1},X_{2},X_{3},\ldots,X_{n} vanishes, but none of its moments ∫ R n ψ ⁢ ( x ) ⁢ x m ⁢ d x \int_{\mathbb{R}^{n}}\psi(x)x^{m}\,dx is zero; here x m = x 1 m 1 ⁢ x 2 m 2 ⁢ … ⁢ x n m n x^{m}=x_{1}^{{m_{1}}}\,x_{2}^{{m_{2}}}\ldots x_{n}^{{m_{n}}} , d ⁢ x = d ⁢ x 1 ⁢ d ⁢ x 2 ⁢ … ⁢ d ⁢ x n dx=dx_{1}\,dx_{2}\ldots dx_{n} and m = ( m 1 , m 2 , … , m n ) m=(m_{1},m_{2},\ldots,m_{n}) and each of m 1 , m 2 , … , m n m_{1},m_{2},\ldots,m_{n} is at least 1. The set of such kernel will be denoted by D m ⁢ ( R n ) \mathcal{D}_{m}(\mathbb{R}^{n}) . But the uniqueness theorem for our wavelet inversion formula is valid for the space D F ′ ⁢ ( R n ) \mathcal{D}_{F}^{\prime}(\mathbb{R}^{n}) obtained by filtering (deleting) (i) all non-zero constant distributions from the space D ′ ⁢ ( R n ) \mathcal{D}^{\prime}(\mathbb{R}^{n}) , (ii) all non-zero constants that appear with a distribution as a union as for example for x 1 2 + x 2 2 + ⋯ ⁢ x n 2 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 = 1 - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}}=1-\frac{1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} , 1 is deleted and - 1 1 + x 1 2 + x 2 2 + ⋯ ⁢ x n 2 \frac{-1}{1+x_{1}^{2}+x_{2}^{2}+\cdots x_{n}^{2}} is retained.

The product arbitrariness of generalised functions and its role in quantum field theory

In this study, the problem of the product arbitrariness of generalised functions in the framework of Schwartz distribution is addressed. This arbitrariness is responsible for the problem of infinities encountered in quantum field theory when higher-order corrections are considered. The methods of Güttinger-König and Hörmander for getting rid of this arbitrariness are investigated.

The Spectra of Principal Submatrices in Rotationally Invariant Hermitian Random Matrices and the Markov– Krein Correspondence

We prove a concentration phenomenon on the empirical eigenvalue distribution (EED) of the principal submatrix in a random hermitian matrix whose distribution is invariant under unitary conjugacy; for example, this class includes GUE (Gaussian Unitary Ensemble) and Wishart matrices. More precisely, if the EED of the whole matrix converges to some deterministic probability measure $\mathfrak{m}$, then its fluctuation from the EED of the principal submatrix, after a rescaling, concentrates at the Rayleigh measure (in general, a Schwartz distribution) associated with $\mathfrak{m}$ by the Markov--Krein correspondence. For the proof, we use the moment method with Weingarten calculus and free probability. At some stage of calculations, the proof requires a relation between the moments of the Rayleigh measure and free cumulants of $\mathfrak{m}$. This formula is more or less known, but we provide a different proof by observing a combinatorial structure of non-crossing partitions.