The resolvent function of an operator in a Banach space is defined on an open subset of the complex plane and is holomorphic. It obeys the resolvent equation. A generalization of this equation to Schwartz distributions is defined and a Schwartz distribution, which satisfies that equation is called a resolvent distribution. In important cases the resolvent distribution is the whole plane. Its restriction to the subset, where it is continuous, is the usual resolvent function. Its complex conjugate derivative is ,but a factor, the spectral Schwartz distribution, which is carried by a subset of the spectral set of the operator. The spectral distribution yields a spectral decomposition. We have a generalized orthogonality relation. Completeness is defined in a natural way and is the case e.g. if the operator is bounded. The spectral distribution of a matrix and a unitary operator are given. If the the operator is a self-adjoint operator on a Hilbert space, the spectral distribution is the derivative of the spectral family. We calculate the spectral distribution and the eigen value problem of the multiplication operator and of the multiplication operator with a non symmetric rank one perturbation. The operator is not normal and may have discrete real or imaginary eigenvalues or a nontrivial Jordan decomposition.
The resolvent of an operator in a Banach space is defined on an open subset of the complex plane and is holomorphic. It obeys the resolvent equation. A generalization of this equation to Schwartz distributions is defined and a Schwartz distribution, which satisfies that equation is called a resolvent distribution. Its restriction to the subset, where it is continuous, is the usual resolvent function. Its complex conjugate derivative is,but a factor, the spectral Schwartz distribution, which is carried by a subset of the spectral set of the operator. The spectral distribution yields a spectral …
In this paper we prove and apply a theorem of spectral expansion for Schwartz linear operators which have an S-linearly independent Schwartz eigenfamily. This type of spectral expansion is the analogous of the spectral expansion for self-adjoint operators of separable Hilbert spaces, but in the case of eigenfamilies of vectors indexed by the real Euclidean spaces. The theorem appears formally identical to the spectral expansion in the finite dimensional case, but for the presence of continuous superpositions instead of finite sums. The Schwartz expansion we present is one possible rigorous and…
Let $X= G/K$ symmetric space of non compact type, where $G$ is a rank-one connected semisimple Lie group with finite center. We shall look at the transform $ P_λf(x) = f \ast φ_λ(x)$, where, $λ\in \mathbb C$ and $φ_λ$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz spaces $S^p(X) $ where $0 < p \leq 2$ under the above transform.
We list multiplier and convolutor spaces of the spaces occurring in L. Schwartz' "Théorie des distributions". Furthermore we clarify whether the multiplications and convolutions are continuous or not.
In this paper we introduce and study the multiplication among smooth functions and Schwartz families. This multiplication is fundamental in the formulation and development of a spectral theory for Schwartz linear operators in distribution spaces, to introduce efficiently the Schwartz eigenfamilies of such operators and to build up a good functional calculus for them. The definition of eigenfamily is absolutely natural and this new operation allows us to develop a rigorous and manageable spectral theory for Quantum Mechanics, since it appears in a form extremely similar to the current use in Ph…
In this paper we define the Schwartz linear operators among spaces of tempered distributions. These operators are the analogous of linear continuous operators among separable Hilbert spaces, but in the case of spaces endowed with Schwartz bases having a continuous index set. The Schwartz linear operators enjoy properties very similar to those enjoyed by linear operators in the finite dimensional case. The Schwartz operators are one possible rigorous mathematical model for the operators and observables used in Quantum Mechanics.
Euler operators are partial differential operators of the form $P(θ)$ where $P$ is a polynomial and $θ_j = x_j \partial/\partial x_j$. They are surjective on the space of temperate distributions on $R^d$. We show that this is, in general, not true for the space of Schwartz distributions on $R^d$, $d\ge 3$, for $d=1$, however, it is true. It is also true for the space of distributions of finite order on $R^d$ and on certain open sets $Ω\subset R^d$, like the euclidian unit ball.
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions $\mathcal{S}'(\mathbb{R}^d)$ and distributions of exponential type $\mathcal{K}'_{1}(\mathbb{R}^{d})$ onto their images.
In this paper we define Schwartz families in tempered distribution spaces and prove many their properties. Schwartz families are the analogous of infinite dimensional matrices of separable Hilbert spaces, but for the Schwartz test function spaces, having elements (functions) realizable as vectors indexed by real Euclidean spaces (ordered families of scalars indexed by real Euclidean spaces). In the paper, indeed, one of the consequences of the principal result (the characterization of summability for Schwartz families) is that the space of linear continuous operators among Schwartz test functi…
Let G be a solvable Lie group endowed with right Haar measure. We define and study a dense Frechet *-subalgebra S of L1(G), consisting of smooth functions rapidly-decreasing at infinity on G. When G is nilpotent, we recover the classical Schwartz algebra introduced by R. Howe and other authors. We develop a distribution theory for S, and we generalize the classical Kernel Theorem of L. Schwartz to our setting.
We construct an algebra of generalized functions $^*\mathcal{E}(\mathbb{R}^d)$. We also construct an embedding of the space of Schwartz distributions $\mathcal{D}^\prime(\mathbb{R}^d)$ into $^*\mathcal{E}(\mathbb{R}^d)$ and thus present a solution of the problem of multiplication of Schwartz distributions which improves J.F. Colombeau's solution. As an application we prove the existence of a weak delta-like solution in ${^*\mathcal{E}(\mathbb{R}^d)}$ of the Hopf equation. This solution does not have a counterpart in the classical theory of partial differential equations. Our result improves a …
Sufficient conditions for existence and uniqueness of the solution of the Volterra integral equations of the first kind with piecewise continuous kernels are derived in framework of Sobolev-Schwartz distribution theory. The asymptotic approximation of the parametric family of generalized solutions is constructed. The method for the solution's regular part refinement is proposed using the successive approximations method.
In this paper we make an attempt to extend L. Schwartz's classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace translations by proper euclidean motions in higher dimensions. For this purpose we introduce the basic concepts of spectral analysis and synthesis in the non-commutative setting based on Gelfand pairs, where "translation invariance" will be replaced by invariance with respect to a compact group of automorphisms. The role of exponential functions will be playe…
We define the spaces of Schwartz functions, tempered functions and tempered distributions on manifolds definable in polynomially bounded o-minimal structures. We show that all the classical properties that these spaces have in the Nash category, as first studied in Fokko du Cloux's work, also hold in this generalized setting. We also show that on manifolds definable in o-minimal structures that are not polynomially bounded, such a theory can not be constructed. We present some possible applications, mainly in representation theory.
Given a $2$-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian $\mathcal{L}$ and a family $\mathcal{T}$ of elements of the derived algebra, we study the convolution kernels associated with the operators of the form $m(\mathcal{L}, -i \mathcal{T})$. Under suitable conditions, we prove that: i) if the convolution kernel of the operator $m(\mathcal{L},-i \mathcal{T})$ belongs to $L^1$, then $m$ equals almost everywhere a continuous function vanishing at $\infty$ (`Riemann-Lebesgue lemma'); ii) if the convolution kernel of the operator …
This paper is a survey of a new family of Banach spaces ${KS}^2$ and $SD^2$ that provide the same structure for the Henstock-Kurzweil (HK) integrable functions as the $L^p$ spaces provide for the Lebesgue integrable functions. These spaces also contain the wide sense Denjoy integrable functions. They were first use to provide the foundations for the Feynman formulation of quantum mechanics. It has recently been observed that these spaces contain the test functions $\mathcal{D}$ as a continuous dense embedding. Thus, by the Hahn-Banach theorem, $\mathcal{D}' \subset \mathcal{B}'$. A new family …
Let $\mathcal{L}$ be a sub-Laplacian on a connected Lie group $G$ of polynomial growth. It is well known that, if $F : \mathbb{R} \to \mathbb{C}$ is in the Schwartz class $\mathcal{S}(\mathbb{R})$, then the convolution kernel $\mathcal{K}_{F(\mathcal{L})}$ of the operator $F(\mathcal{L})$ is in the Schwartz class $\mathcal{S}(G)$. Here we prove a sort of converse implication for a class of groups $G$ including all solvable noncompact groups of polynomial growth. We also discuss the problem whether integrability of $\mathcal{K}_{F(\mathcal{L})}$ implies continuity of $F$.
One can represent Schwartz distributions with values in a vector bundle $E$ by smooth sections of $E$ with distributional coefficients. Moreover, any linear continuous operator which maps $E$-valued distributions to smooth sections of another vector bundle $F$ can be represented by sections of the external tensor product $E^* \boxtimes F$ with coefficients in the space $\mathcal{L}(\mathcal{D}', C^\infty)$ of operators from scalar distributions to scalar smooth functions. We establish these isomorphisms topologically, i.e., in the category of locally convex modules, using category theoretic fo…
We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, an…
In this paper we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them in particular with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Frechet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show in particular that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some u…
We tackle the problem of finding a suitable categorical framework for generalized functions used in mathematical physics for linear and non-linear PDEs. We are looking for a Cartesian closed category which contains both Schwartz distributions and Colombeau generalized functions as natural objects. We study Frölicher spaces, diffeological spaces and functionally generated spaces as frameworks for generalized functions. The latter are similar to Frölicher spaces, but starting from locally defined functionals. Functionally generated spaces strictly lie between Frölicher spaces and diffeological s…
Given a graded group $G$ and commuting, formally self-adjoint, left-invariant, homogeneous differential operators $\mathcal{L}_1,\dots, \mathcal{L}_n$ on $G$, one of which is Rockland, we study the convolution operators $m(\mathcal{L}_1,\dots, \mathcal{L}_n)$ and their convolution kernels, with particular reference to the case in which $G$ is abelian and $n=1$, and the case in which $G$ is a $2$-step stratified group which satisfies a slight strengthening of the Moore-Wolf condition and $\mathcal{L}_1,\dots,\mathcal{L}_n$ are either sub-Laplacians or central elements of the Lie algebra of $G$.…
The Schwartz kernel of the spectral density for the Schrödinger operator with magnetic field in the $n-$dimensional complex ball is given. As applications, we compute the heat, resolvent and the wave kernels. Moreover, the resolvent and wave kernels are used to establish two new formulas for the Gauss-hypergeometric function.
We investigate the $τ$-quantizations and Cohen's class distributions of a suitable class of trace-class operators, called Feichtinger's operators, and show that it is a convenient substitute for the class of Schwartz operators. Many well-known concepts and results for functions in time-frequency analysis have an operator-analog in our setting, e.g. that Cohen's classes are convolutions of Wigner functions with distributions or characterization of the class of Schwartz operators as an intersection of weighted variants of the class of Feichtinger operators.
We characterize Schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by A. Robinson. They appear to be distributions continuous in a neighborhood of the point.
We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. We show that the various constructions of such embeddings existing in the literature lead in fact to the same one.
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of {\em Colombeau type} in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. The article is aimed at mathematicians and physicists who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theor…
As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the present paper we construct a space of distributions where the general operation of multiplication by a discontinuous function is defined, continuous, commutative, associative and for which the Leibniz product rule holds. In the new space of distributions, the classical delta-function $δ_τ$ extends to a family of delta-functions $δ_τ^α$, dependent on the \te…
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers w…
We apply L.~Schwartz' theory of vector valued distributions in order to simplify, unify and generalize statements about convolvability of distributions, their regularization properties and topological properties of sets of distributions. The proofs rely on propositions on the multiplication of vector-valued distributions and on the characterization of the spaces $\mathcal{O}_{M}(E,F)$ and $\mathcal{O}_{C}'(E,F)$ of multipliers and convolutors for distribution spaces $E$ and $F$.
We calculate crossing probabilities and one-sided last exit time densities for a class of moving barriers on an interval $[0,T]$ via Schwartz distributions. We derive crossing probabilities and first hitting time densities for another class of barriers on $[0,T]$ by proving a Schwartz distribution version of the method of images. Analytic expressions for crossing probabilities and related densities are given for new explicit and semi-explicit barriers.
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau's solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn-Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although ind…
The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Four…
We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert-Schmidt operator. The main result is applied to Cohen's class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen's class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert-Schmidt operator must satisfy a nuclearity condition for these results to hold, and we…
We study regularizations of Schwartz distributions on a complete Riemannian manifold $M$. These approximations are based on families of smoothing operators obtained from the solution operator to the wave equation on $M$ derived from the metric Laplacian. The resulting global regularization processes are optimal in the sense that they preserve the microlocal structure of distributions, commute with isometries and provide sheaf embeddings into algebras of generalized functions on $M$.
The aim of this paper is to prove that the well known non solvable Mizohata type partial differential equations have Colombeau generalized solutions which are distributions if and only if they are solv- able in the space of Schwartz distributions. Therefore the Colombeau generalized solvability includes both a new solution concept and new mathematical objects as solutions.
The aim of this paper is to develop the theory of distributions, not necessarily of compact support, in a topos model of Synthetic Differential Geometry, the so-called "Cahiers Topos". As an application, we study the evolution through time of a heat distribution. In the first part of the paper we study distributions in the sense of Schwartz in the context of convenient vector spaces. In particular, we study smoothness (with respect to time) of the fundamental distribution solution of the heat equation. In the second part, we show how the results of the first part may be used to internalize the…
A concise derivation of a new multiplicative product of Schwartz distributions is presented. The new product $\star$ is defined in the vector space ${\cal A}$ of piecewise smooth functions on $\bkR$ and all their (distributional) derivatives; it is associative, satisfies the Leibnitz rule and reproduces the usual product of functions for regular distributions. The algebra $({\cal A},+,\star)$ yields a sufficiently general setting to address some interesting problems. As an application we consider the problem of deriving a global formulation for quantum confined systems.
We study an infinite dimensional analysis with respect to the measure on Schwartz space of tempered distributions, corresponding to the distributional derivative of gamma process. Laguerre polynomials being orthogonal with respect to gamma noise measure turn out to be generalized Appell ones. This fact enables to generalize the white noise functional approach on the stochastic Wick-Skorokhod equations involving gamma noise. E. g. we consider Werhulst type equation driven by gamma noise.
