No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Topological Vector Spaces and Distributions I
... The family w p (E) (or c w 0 (E) if p = ∞) of all weakly p-summable sequences in E is a vector space which admits a natural locally convex vector topology such that it is complete if so is E; for details, see Section 19.4 in [27] or Section 4 in [17]. Analogously, we say that a sequence {χ n } n∈ω in E is weak * p-summable if ( χ n , x ) ∈ p (or ( χ n , x ) ∈ c 0 if p = ∞) for every x ∈ E. For the basic theory of locally convex spaces, see the classical books [26,27,32,34]. ...
... Therefore, in both cases (i) and (ii) we can assume that S is equicontinuous. Hence, by Proposition 3.9.8 of [26], the weak * topology σ(E , E) and the topology τ pc of uniform convergence on precompact subsets of E coincide on S . Since S is weak * p-summable, it is a weak * null-sequence. ...
... The weak * p-summability of S implies that S is weak * null. Hence, by Proposition 3.9.8 of [26], ...
Let Being motivated by the classical notions of limited, p -limited, and coarse p -limited subsets of a Banach space, we introduce and study -limited subsets and their equicontinuous versions and coarse p -limited subsets of an arbitrary locally convex space E . Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact, and relatively weakly sequentially (pre)compact sets. If E is a Banach space, we show that the class of coarse 1 -limited subsets of E coincides with the class of -limited sets, and if , then the class of coarse p -limited sets in E coincides with the class of p - sets of Pełczyński. We also generalize a known theorem of Grothendieck.
... 11 As any other compatible with duality such as the strong or the McKey topologies. 12 The antilinearity is obvious. To show continuity, let us choose an arbitrary φ ∈ Φ. ...
... The space Φ is a locally convex space [12], with a topology defined by a set of seminorms 8 of which one has to be the Hilbert space norm defined on Φ as a subspace of H. In our particular context, the topology on Φ will be defined by a countably infinite set of seminorms, so that the space Φ be metrizable 9 . ...
... Since ψ(x) ∈ S, the coefficients {a n } must satisfy (12). Let us define aψ, the action of the operator a on ψ(x) as 6 of 24 ...
In the present paper and inspired with a similar construction on Hermite functions, we construct two series of Gelfand triplets each one spanned by Laguerre-Gauss functions with a fixed positive value of one of their parameters, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre-Gauss functions with negative value of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.
... Con lo cual, tenemos la intersección disjunta deseada. □ Cabe resaltar que algunos autores como [Conway, 1985] o [Horváth, 2012] no imponen ninguna condición adicional a un TVS. En [Rudin, 1991], tenemos la condición: cada singleton {x} ⊂ X es cerrado. ...
... Base local es también llamado sistema fundamental de vecindades [Horváth, 2012] pues en realidad una base local es una base del filtro de vecindades del origen (pues la colección de las vecindades de un punto en cualquier espacio topológico es un filtro), de allí el nombre de fundamental. ...
... Teorema 3.3 [Horváth, 2012] En cualquier espacio topológico, existe una base local B tal que: ...
We have undertaken the development of the functional theory of Schwartz space, commonly referred to as the Space of Rapidly Decreasing functions. In doing so, we establish a topology, generated by a family of separable seminorms, that imparts the Schwartz space with a multitude of significant properties. These properties include its classification as a Polish space, a Frechet space, a Montel space, and a space where the Fourier transform becomes a continuous linear automorphism.
... L.Schwartz publicó cuatro articulas sobre su teoría de distribuciones antes de publicar su famoso libro "Théorie des Distributions", [11], libro que marcó una etapa en la evolución de las EDP pues fue una referencia obligada de los investigadores de las EDP. Años después surgieron algunos libros sobre este tema, hoy ya clásicos, como fueron Hormander [1], Traves [14], Horvath [15], Nachbin [16], entre otros. Luego de la publicación de su libro, Schwartz tuvo un gran suceso cuando prueba su "teorema del núcleo"(1952) resultado que le permitió extender su teoría a las distribuciones de valor vectorial. ...
... Es oportuno remarcar que la noción de distribución, según Schwartz es basado en la dualidad, en la teoría de los espacios vectoriales topológicos, ver [15] y [19]; así el espacio D ′ (R n ) consiste de todas las funcionales continuas sobre C ∞ 0 (R n ), esto es,es el espacio dual del espacio de las funciones test D, y está provisto de la topología que contiene la convergencia de derivadas de todas lasórdenes. Así, entonces, toda distribución T puede ser representada localmente como una suma finita de derivadas, en el sentido de las distribuciones, de funciones continuas. ...
Partial differential equations (PDE’s) are, we believe, little known in our country, especially at a medium-advanced level, in particular, their historical roots and methods developed in their evolution. The objective of this article is to contribute to a knowledge in this direction especially to colleagues and students who are interested in studying and researching in this central Branch of mathematics. As a model we have chosen H¨ormander’s book, [1], whose study and teaching will be a sign of great progress in this direction in our country. In this article we only give an introduction to part I dedicated to functional analysis and which includes chapters I and II that deal with the theory of distributions and some special spaces of distributions, respectively.
... A complete metrizable locally convex space (A locally convex space is a topological vector space such that the origin has a fundamental system of convex neighborhoods. All spaces that appear along the present article are locally convex [3]) is a Frèchet space (Here, norms could be replaced by seminorms. The difference between a norm and a seminorm is that the seminorm of a vector could be zero even if this vector is not zero. ...
... The second topology on Φ to be considered here is a strict inductive limit of metrizable spaces [3], to be described later. Now, let ϕ be an arbitrary vector in the Hilbert space H. ...
Rigged Hilbert spaces (RHSs) are the right mathematical context that include many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS, discrete and continuous bases, as well as an abstract basis and the basis of special functions and representations of Lie algebras of symmetries are used by continuous operators. This is not possible in Hilbert spaces. In the present paper, we study a model showing all these features, based on the one-dimensional Pöschl–Teller Hamiltonian. Also, RHS supports representations of all kinds of ladder operators as continuous mappings. We give an interesting example based on one-dimensional Hamiltonians with an infinite chain of SUSY partners, in which the factorization of Hamiltonians by continuous operators on RHS plays a crucial role.
... As lim ε→0 ρ ε = 1, we get lim ε→0 ( u ρ ε ) ε ∈ O M R d . (For those limits, we consider O M R d equipped with its usual topology: See [16], [24].) This shows the consistency of our result with the classical one. ...
... We turn to the question of embeddings. First, the structure of elements of O ′ C R d ( [16], [21], [24] ...
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.
... A Fréchet space is in our terminology a locally convex metrizable complete topological vector space. For LF-spaces, i.e., inductive limits of Fréchet spaces, we refer to [31,14,28]. ...
... When C m c (Ω) ′ and C m (Ω) ′ carry their weak topologies, C ∞ c (Ω) is in both cases a sequentially dense subspace. We refer to [31,14] for details. We write D for C ∞ c (R n ) and D ′ for the distributions on R n . ...
We take a unifying and new approach toward polynomial and trigonometric approximation in an arbitrary number of variables, resulting in a precise and general ready-to-use tool that anyone can easily apply in new situations of interest. The key idea is to show, in considerable generality, that a module, which is generated over the polynomials or trigonometric functions by some set, necessarily has the same closure as the module which is generated by this same set, but now over the compactly supported smooth functions. The particular properties of the ambient space or generating set are to a large degree irrelevant. This translation -- which goes in fact beyond modules -- allows us, by what is now essentially a straightforward check of a few properties, to replace many classical results by more general and stronger statements of a hitherto unknown type. As a side result, we also obtain a new integral criterion for multidimensional measures to be determinate. At the technical level, we use quasi-analytic classes in several variables and we show that two well-known families of one-dimensional weights are essentially equal. The method can be formulated for Lie groups and this interpretation shows that many classical approximation theorems are "actually" theorems on the unitary dual of n-dimensional real space. Polynomials then correspond to the universal enveloping algebra.
... Then E ′ = n∈ω U • n , where U • n is σ(E ′ , E)-compact by the Alaoglu theorem. Hence U • is τ k -compact by Proposition 3.9.8 of [13]. By (3) of [15, §21.10], τ k coincides with the precompact topology τ pc on E ′ . ...
... Proof. By the Alaoglu theorem, the polar U • is σ(E ′ , E)-compact, and hence U • is a τ pc -compact disc by Proposition 3.9.8 of [13]. Therefore the T -compact set U • is not absorbing (otherwise, U • would be a compact neighborhood of zero in (E ′ , T ) by the barrelledness of (E ′ , T ), and hence E is finite-dimensional). ...
Following [2], a Tychonoff space X is Ascoli if every compact subset of is equicontinuous. By the classical Ascoli theorem every k-space is Ascoli. We show that a strict (LF)-space E is Ascoli iff E is a Fr\'{e}chet space or . We prove that the strong dual of a Montel strict (LF)-space E is an Ascoli space iff one of the following assertions holds: (i) E is a Fr\'{e}chet--Montel space, so is a sequential non-Fr\'{e}chet--Urysohn space, or (ii) , so . Consequently, the space of test functions and the space of distributions are not Ascoli that strengthens results of Shirai [20] and Dudley [5], respectively.
... From Lemma 2 above, we may consider the space E (R) equipped with the topology arising from the family of seminorms Γ k,K . From ( [11], Proposition 2, p.97), there exist C > 0, a compact set K ⊂ R, and a non-negative integer q, all depending on f , such that ...
... Therefore, from Theorem 1 and relation (11), one has the following. ...
This paper explores Abelian theorems associated with the real Weierstrass transform over distributions of compact support. This study contributes to both mathematical analysis and distribution theory by offering new insights into the interaction between integral transforms and compactly supported distributions.
... for ϕ ∈ D. This is the most common definition of convolution of distributions, which can be found up to details in [116,117]. For example, convolvability by support is guaranteed, ...
... When X is a normed space, then β(X , X ) is the usual dual norm topology. [5], page# [117], page# [161], page# B c ( ) is defined as the space B( ) equipped with the finest locally convex topology that induces the topology of E( ) on the subsets of B( ) which are bounded w.r.t its Frechet topology generated by the seminorms ϕ → ∂ κ ϕ ∞ , κ ∈ N d 0 . This topology is weaker than the Frechet topology on B( ) = D L ∞ ( ). ...
Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as "Equation missing" -convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of "Equation missing" -convolution.
... Lemma 1.19: [7] Let X be a vector space, ∅AX. For xA ...
In this work some properties of semi-pre irresolute topological vector space was introduced , also several characterizations of semi-pre Hausdorff are given. Moreover, we show that the extreme point of convex subset of semi-pre irresolute topological space X lies on the boundary. مترددة الشبه المتجهه التبولوجيه ألفضاءات علي محمد ابراهيم راضي ، البياتي حسين حاتم جالل و حميد كريم سهاد الرياضيات قسم-للبنات العلوم كليه-بغداد جامعه ألخالصة , خصائصها دراسة و مترددة الشبه المتجهه التبولوجيه الفضاءات تعريف تم البحث هذا في للفضاءات تمثيل وأعطاء (هاوزدورف الشبه semi-pre Hausdorff المحدبه الجزئيه للمجموعات القصوى القيم ان اثبات تم ذلك الى أضافة .). مترددة الشبه المتجهه التبولوجيه للفضاءات بالنسبه حدوديه تكون المفتاحيه الكلمات المتجهه التبولوجيه الفضاءات : ، التبولوجيه الفضاءات مترددة الشبه المتجهه ، شبه الفضاءات هاوزدورف ،. القصوى القيم
... Lemma 1.19: [7] Let X be a vector space, ∅AX. For xA ...
... From Lemma 2 above, we may consider the space E (R + ) equipped with the topology arising from the family of seminorms Γ k,K . From [21] [Proposition 2, p. 97], there exist C > 0, a compact set K ⊂ R + , and a non-negative integer q, all depending on f , such that ...
The objective of this research is to obtain the asymptotic behaviour of compactly supported distributions and generalized functions for a variant of the index Whittaker transform.
... where the derivatives f (n) are given in the distributional sense (see [12]). Furthermore, we may introduce the notion of multiplicator χ , which is a particular tempered function belonging to C ∞ (R), such that χ and all its derivatives are tempered (see, e.g., [39,47,52]). In this way, if γ ∈ S then the product χγ ∈ S and the following distribution turns out to be well-defined ...
In this paper, we are concerned with the study of the regularization properties of Durrmeyer-sampling type operators in -spaces, with . In order to reach the above results, we mainly use tools belonging to distribution theory and Fourier analysis. Here, we show how the regularization process performed by the operators is strongly influenced by the regularity of the discrete kernel . We investigate the classical case of continuous kernels, the more general case of kernels in Sobolev spaces, as well as the remarkable case of bandlimited kernels, i.e., belonging to Bernstein classes. In the latter case, we also establish a closed form for the distributional Fourier transform of the above operators applied to bandlimited functions. Finally, the main results presented herein will be also applied to specific instances of bandlimited kernels, such as de la Vallée Poussin and Bochner–Riesz kernels.
... We review some facts and definitions on Hermite functions and on dual of nuclear Fréchet spaces; see e.g. [17,30] for more information on the topic. ...
With the use of Hida's white noise space theory space theory and spaces of stochastic distributions, we present a detailed analytic continuation theory for classes of Gaussian processes, with focus here on Brownian motion. For the latter, we prove and make use a priori bounds, in the complex plane, for the Hermite functions; as well as a new approach to stochastic distributions. This in turn allows us to present an explicit formula for an analytically continued white noise process, realized this way in complex domain. With the use of the Wick product, we then apply our complex white noise analysis in a derivation of a new realization of Hilbert space-valued stochastic integrals
... We begin with two definitions: Definition 1 (Dual system). Let X and Y be finite m-dimensional vector spaces [23] equipped with a pairing product b(·, ·), i.e., a bilinear map: ...
We present a generalization of Bregman divergences in finite-dimensional symplectic vector spaces that we term symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel–Young inequality which relies on the notion of symplectic subdifferentials. The symplectic Fenchel–Young inequality is obtained using the symplectic Fenchel transform which is defined with respect to the symplectic form. Since symplectic forms can be built generically from pairings of dual systems, we obtain a generalization of Bregman divergences in dual systems obtained by equivalent symplectic Bregman divergences. In particular, when the symplectic form is derived from an inner product, we show that the corresponding symplectic Bregman divergences amount to ordinary Bregman divergences with respect to composite inner products. Some potential applications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are touched upon.
... Then the c 0 -barrelledness of E implies that the sequence S = {χ n : n ∈ N}∪ {0} is equicontinuous. So S ⊆ U • for some neighborhood U of zero in E. Since σ(E ′ , E)| U • = τ k | U • by Proposition 9.3.8 of [28], we obtain σ(E ′ , E)| U • = µ(E ′ , E)| U • . Thus χ n → 0 in µ(E ′ , E). ...
In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and have the same sequentially compact sets, where is the space E with the weak topology, (3) E and have the same compact sets, (4) E and have the same countably compact sets, (5) E and have the same pseudocompact sets, (6) E and have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in .
... div h a pxq " 0 for x P R d zto, au. (9) and, moreover, by (6) we get (10) If U is a bounded convex domain containing to, au, then ż BU xh a pxq, νpxqy dH n´1 pxq " 0. ...
Let E be a finite-dimensional normed space and a nonempty convex open set in E. We show that the Lipschitz-free space of is canonically isometric to the quotient of by the subspace consisting of vector fields with zero divergence in the sense of distributions on E.
... 3.1. We recall here (see [22], [29], or [30]) some wellknown results from the theory of distributions which we shall need in the following. We consider the following spaces of smooth functions on R n : ...
Looking for the universal covering of the smooth non-commutative torus leads to a curve of associative multiplications on the space \Cal O_M'(\Bbb R^{2n})\cong \Cal O_C(\Bbb R^{2n}) of Laurent Schwartz which is smooth in the deformation parameter . The Taylor expansion in leads to the formal Moyal star product. The non-commutative torus and this version of the Heisenberg plane are examples of smooth *-algebras: smooth in the sense of having many derivations. A tentative definition of this concept is given.
... The space Φ is a locally convex space [19], with a topology defined by a set of seminorms, (A seminorm is a mapping p : ...
Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative values of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.
... Definition 1 (Dual system.) Let X and Y be topological locally convex m-dimensional vector spaces [21] (LCTVSs) equipped with a pairing product b(·, ·), i.e., a bilinear map: ...
We present a generalization of Bregman divergences in symplectic vector spaces called symplectic Bregman divergences. Symplectic Bregman divergences are derived from a symplectic generalization of the Fenchel-Young inequalities which rely on symplectic subdifferentials. The generic symplectic Fenchel-Young inequality is obtained using symplectic Fenchel transforms which are defined with respect to linear symplectic forms. Some potential appplications of symplectic divergences in geometric mechanics, information geometry, and learning dynamics in machine learning are discussed.
... This shows that φ is weak* continuous on bounded subsets of E * , as desired. By [9,Corollary 4], φ is weak* continuous on E * and thus there is a ∈ E such that φ(ψ) = ψ(a) for all ψ ∈ E * . Now, take an n-homogenous nuclear polynomial P = ∑ ∞ i=1 ψ n i with ψ i ∈ E * and ∑ ∞ i=1 ψ i n < ∞. ...
Let E be a Banach space and A be a commutative Banach algebra with identity. Let be the space of A-valued polynomials on E generated by bounded linear operators (an n-homogenous polynomial in is of the form , where , , are bounded linear operators and ). For a compact set K in E, we let be the closure in of the restrictions of polynomials P in . It is proved that is an A-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product , where is the uniform algebra on K generated by nuclear scalar-valued polynomials. The character space of is then identified with where is the nuclear polynomially convex hull of K in E, and is the character space of A.
... so the claim follows from [39,Ch. 4,Prop. ...
We study rigidity problems for Riemannian and semi-Riemannian manifolds with metrics of low regularity. Specifically, we prove a version of the Cheeger-Gromoll splitting theorem \cite{CheegerGromoll72splitting} for Riemannian metrics and the flatness criterion for semi-Riemannian metrics of regularity . With our proof of the splitting theorem, we are able to obtain an isometry of higher regularity than the Lipschitz regularity guaranteed by the -splitting theorem \cite{gigli2013splitting, gigli2014splitoverview}. Along the way, we establish a Bochner-Weitzenb\"ock identity which permits %incorporates both the non-smoothness of the metric and of the vector fields, complementing a recent similar result in \cite{mondino2024equivalence}. The last section of the article is dedicated to the discussion of various notions of Sobolev spaces in low regularity, as well as an alternative proof of the equivalence (see \cite{mondino2024equivalence}) between distributional Ricci curvature bounds and -type bounds, using in part the stability of the variable -condition under suitable limits \cite{ketterer2017variableCD}.
... The mathematical pre-context of our analysis is, so, the Laurent Schwartz distribution theory, in particular, that based on the Minkowski space-time. For distribution theory, we refer to Schwartz [29][30][31][32][33][34], Barros-Neto [1], Dieudonné [23,24], Bourbaki [3][4][5], Horváth [26], Kesavan [27], Boccara [2], Lang [28], Trèves [36], Yosida [37] and Zeidler [38]. ...
In this work, we start from the problem of quantizing the relativistic Hamiltonian of a free massive particle (rest mass different from 0), a problem exceptionally difficult in the standard approaches to quantum mechanics. In fact, in tempered distribution state space, we find the natural way to define the relativistic Hamiltonian operator and its associated Schrödinger equation The existence of a linear continuous Hermitian operator associated with the Einstein's Hamiltonian of a free particle, defined on the entire tempered distribution space, automatically implies the conservation of Born probability flux (which doesn't mean the conservation of particles number, rather it implies the conservation of the total relativistic energy of the solution wave). We, then, deduce the continuity equation for the Born probability density and study some its different (but equivalent) expressions. We determine some possible forms of probability currents and flux velocity fields associated with the particle evolution. We provide the relativistic invariant expression for both Schrödinger equation and probability flux continuity equations.
... We first collect results from the theory of distributions that are needed for the treatment of general lattice sums. We begin with the definition of the spaces involved [31]. Definition 6.4 (Translations). ...
This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with 3×10^23 particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
... That is, the jth row ∇g j = (D 1 g j , ..., D n g j ) of ∇g equals the jth rowσ = (σ j1 , ...,σ jn ) ofσ in S ′ (R n , R n ). See [56,Chapter 4, Section 3, Proposition 9] of [85, Chapter II, Section 6, Théorème VI] (the result is stated for the dual of smooth compactly supported functions, but the proof does not change for Schwartz functions and tempered distributions). For any j and k we have D k g j =σ jk ∈ L 1 (R n ) ∩ L ∞ (R n ) by Assumption 3.12 and Lemma 3.14. ...
... The mathematical precontext of our analysis is the Laurent Schwartz distribution theoryin particular, that based on the Minkowski space-time. For distribution theory, we refer to Schwartz [1][2][3][4][5][6], Barros-Neto [7], Dieudonné [8,9], Bourbaki [10][11][12], Horváth [13], Kesavan [14], Boccara [15], Lang [16], Trèves [17], Yosida [18] and Zeidler [19]. ...
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.
The present article presents geometric quantization on cotangent bundles as a special instance of Kirillov's orbit method. To this end, the cotangent bundle is realized as a coadjoint orbit of an infinite-dimensional Lie group constructed from the diffeomorphism group. We also develop two asymptotic character formulas by employing the tangent groupoid.
We develop criteria to guarantee uniqueness of the C∗‐norm on a ∗‐algebra B. Nontrivial examples are provided by the noncommutative algebras of C‐valued functions SJC(Rn) and BJC(Rn) defined by M.A. Rieffel via a deformation quantization procedure, where C is a C∗‐algebra and JJ is a skew‐symmetric linear transformation on Rn with respect to which the usual pointwise product is deformed. In the process, we prove that the Fréchet ∗‐algebra topology of BJC(Rn) can be generated by a sequence of submultiplicative ∗‐norms and that, if C is unital, this algebra is closed under the C∞‐functional calculus of its C∗‐completion. We also show that the algebras SJC(Rn) and BJC(Rn) are spectrally invariant in their respective C∗‐completions, when C is unital. As a corollary of our results, we obtain simple proofs of certain estimates in BJC(Rn).
In this paper, we propose a rational and almost obliged path leading us from the most natural assumption regarding the Schrödinger wave functions (wave functions classically belonging to the domain of the classic Schrödinger equation) to the space of tempered distributions defined upon the Minkowski space time. It's extremely natural to consider the space E of complex valued smooth functions as the natural domain of the Schrödinger equation, as it was the obvious implicit assumption of any partial differential equation with constant coefficients at the time of Schrödinger himself. The orthodox Born interpretation of the normalized wave functions and the intervention of von Neumann (a Hilbert's pupil) have deviated the straightforward understanding of that domain towards the unnecessary and unnatural Hilbert space L 2. The Hilbert space of square integrable functions is clearly incompatible with any partial differential equation, which would require at least Sobolev Spaces to live in; unfortunately, the inner product of the Sobolev space H 2 is not good for quantum mechanics and the de Broglie solutions of the Schrödinger equation do not belong to L 2. We show in this article that moving inside the very good space E-and finally inside the huge tempered distribution space S ′-represents the first framework in which any desirable property of the key characters and actors of basic Quantum Mechanics is satisfied.
This paper derives asymptotic behaviours of
Laplace transforms of generalized functions. The results
demonstrate how distribution theory contributes to the depth
of mathematical analysis.
In this elementary but original introduction to the close relationship between the quantum mechanics in complex tempered distribution space and Maxwell’s electromagnetism, we start from an elementary (but representative) case. We prove that, in the chosen particular case, the Maxwell’s equations in empty space can be reformulated as a unique equation inside a subspace of solenoidal smooth vector fields. Moreover, we reinterpret the curl operator as the magnitude of the wave-vector operator. The unique complex equation condensing the two curl Maxwell’s equations reveals to be a faithful representation of the quantization of the Einstein’s energy relation for photons, or, if preferred, the relativistic SchrÅNodinger equation for a massless particle.
This paper is devoted to the concept of Green families associated with a linear continuous endomorphism defined on the space of tempered distributions S'n. Our concept of Green family is an operative and rigorous version (counterpart) of some formal type of Green function used in Theoretical Physics, Quantum Mechanics, Quantum Chemistry and Engineering and sometime also in Mathematical Physics to solve some kind of Linear Differential Equations. The relationships between our method and the approach by fundamental solutions is immediately showed, to solve inhomogeneous linear equations.
The Fourier transform is introduced, and a pointwise inversion theorem for classical functions is proven. The Fourier transform of tempered distributions is then established. Calculations with Fourier transforms are derived with the corresponding rules regarding symmetries, derivatives, integrals, and convolutions. Typical application examples are generalized Fourier series and impulse sequences, polynomials, and pseudofunctions such as rational functions. Important examples for discrete signal processing are also convolutions of impulse sequences with suitable growth properties of their pulse strengths. The Fourier transforms for square-integrable functions and for functions or distributions with several variables are dealt with in separate sections. Fraunhofer diffraction on rectangular and circular apertures is one of the examples. Further examples on all topics can be found in the text and in the exercises of the chapter.
The fundamentals of distribution theory are developed. The Dirac impulse is introduced motivated with a circuit that causes a derivation of an input signal. Starting from this example, the space of distributions is defined and examples of its elements are given. Such elements are, for example, all locally integrable functions, the principal value, and other pseudofunctions like rational functions or 1/—t—. The calculus of distributions is developed to the extent as necessary in the further text. This includes generalized derivatives and convolution of distributions. The results are generalized for multidimensional parameters and test functions over the complex scalar field. Examples for every topic and exercises complete the chapter.
In the theory of inner and outer balayage of positive Radon measures on a locally compact space X to arbitrary with respect to suitable, quite general function kernels, developed in a series of the author’s recent papers, we find conditions ensuring the validity of the integral representations. The results thereby obtained do hold and seem to be largely new even for several interesting kernels in classical and modern potential theory, which looks promising for possible applications. As an example of such applications, we analyze how the total mass of a measure varies under its balayage with respect to fractional Green kernels.
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 and of multipliers and convolutors for distribution spaces E and F.
Analogous to L.~Schwartz' study of the space of semi-regular distributions we investigate the topological properties of the space of semi-regular vanishing distributions and give representations of its dual and of the scalar product with this dual. In order to determine the dual of the space of semi-regular vanishing distributions we generalize and modify a result of A. Grothendieck on the duals of if E and F are quasi-complete and F is not necessarily semi-reflexive.
We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type and Roumieu type are non-trivial if and only if for all and some , respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.
The aim of this workshop was to convene experts for fostering the discussion and the development of innovative approaches in insurance and financial mathematics. New challenges like price instability, huge insurance claims and climate change are affecting the markets, while at the same time the possibility of using large volumes of data and continuously increasing computer power as well as recently developed mathematical methods offer new opportunities for modelling and risk assessment. Here we present an overview of these recent developments by providing the abstracts of the talks that were given during the week, together with a brief summary of the covered topics.
We give a new proof of the completeness of the space by applying a criterion of compact regularity for the isomorphic sequence space . Along the way we show that the strong dual of any quasinormable Fr\'echet space is a compactly regular -space. Finally, we prove that if the inductive limit is compactly regular.
We apply Poisson formula for a strip to give a representation of Z(t) by means of an integral. After that we get the estimate with We explain how the study of this function can lead to information about the zeros of the zeta function on the critical line.
For , and under appropriate conditions on the sequence of weights, the elements, the (weakly, weakly*, strongly) bounded subsets, and the (weakly, weakly*, strongly) convergent sequences in the dual of a space of type Hankel- can be represented by distributional derivatives of functions and measures in terms of iterated adjoints of the differential operator and the Bessel operator . In this paper, such representations are compiled, and new ones involving adjoints of suitable iterations of the Zemanian differential operator are proved. Prior to this, new descriptions of the topology of the space are given in terms of the latter iterations.
We study the problem of existence of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space X. Then we turn to the case that is a space of scalar-valued functions on a non-empty set and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there are a locally convex Hausdorff space Y, a map and a topological isomorphism such that for all .
We show that Shannon's reconstruction formula can be written as a * (b · c) = c = (a * b) · c with tempered distributions a, b, c ∈ S(R^n) where * is convolution, · is multiplication, c is the function being sampled and restored after sampling, b· is sampling and a * its inverse. The requirement a * b = 1 which describes a smooth partition of unity where b = III is the Dirac comb implies that a is satisfied by unitary functions introduced by Lighthill (1958). They form convolution inverses of the Dirac comb. Choosing a = sinc yields Shannon's reconstruction formula where the requirement a * b = 1 is met approximately and cannot be exact because sinc is not integrable. In contrast, unitary functions satisfy this requirement exactly and stand for the set of functions which solve the problem of inverse sampling algebraically.
We propose a formulation of Gamow states, which is the part of unstable quantum states that decays exponentially, with two advantages in relation with the usual formulation of the same concept using Gamow vectors. The first advantage is that this formulation shows that Gamow states cannot be pure states, so that they may have a non-zero entropy. The second is thepossibility of correctly defining averages of observables on Gamow states.
We study the problem of existence and uniqueness of isometric Banach preduals of a Banach space. We derive necessary and sufficient conditions for the existence of an isometric Banach predual of a Banach space X. Then we focus on the case that is a Banach space of scalar-valued functions on a non-empty set and describe those spaces which admit a special isometric Banach predual, namely a strong isometric Banach linearisation, i.e. there is a Banach space Y, a map and an isometric isomorphism such that for all . Finally, we give necessary and sufficient conditions for Banach spaces with a strong isometric Banach linearisation to have a (strongly) unique isometric Banach predual.
In this chapter we introduce the concept of Degree of Nondensifiability
(DND for short) of a bounded set of a Banach space as a measure of nonfilling, i.e. the
quantification of not being filled by a Peano continuum (these sets have a DND equal
to 0). The DND is related with the measures of noncompacteness (MNC for short)
although it is not one of them. However the DND generates an MNC called Degree of
Convex Non-densifiability (DCND for short) which dominates all MNCs. The theory
that is deduced from such a concept provides basic results on the existence of a fixed
point. We also wish to highlight some applications of the Degree of Nondensifiabilty
to the Integral Equations Theory, Operator Theory and to find a solution for a type
of Cauchy-Kovalevskaya problem in a locally convex space defined as a projective
limit of the family of Banach spaces of a generalized scale.
ResearchGate has not been able to resolve any references for this publication.