scholarly journals An embedding of Schwartz distributions in the algebra of asymptotic functions

1998 ◽  
Vol 21 (3) ◽  
pp. 417-428 ◽  
Author(s):  
Michael Oberguggenberger ◽  
Todor Todorov
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions ◽  
Asymptotic Functions ◽  
Space Of Distributions ◽  
Asymptotic Function

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper “asymptotic function,” similar to but different from J. F Colombeau's algebras of new generalized functions.

scholarly journals On Products of Distributions in Colombeau Algebra

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Marija Miteva ◽  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Schwartz Distributions ◽  
Products Of Distributions ◽  
Renormalization Theory

Results on products of distributionsx+-kandδ(p)(x)are derived. They are obtained in Colombeau differential algebra𝒢(R)of generalized functions that contains the space𝒟'(R)of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

scholarly journals Further Results on Colombeau Product of Distributions

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Biljana Jolevska-Tuneska ◽  
Tatjana Atanasova-Pacemska
Keyword(s):  
Differential Algebra ◽  
Generalized Functions ◽  
Schwartz Distributions

Results on Colombeau product of distributions and are derived. They are obtained in Colombeau differential algebra of generalized functions that contains the space of Schwartz distributions as a subspace and has a notion of “association” that is a faithful generalization of the weak equality in .

scholarly journals INTERACTION MEASURES ON THE SPACE OF DISTRIBUTIONS OVER THE FIELD OF p-ADIC NUMBERS

2003 ◽  
Vol 06 (03) ◽  
pp. 389-411 ◽  
Author(s):  
ANATOLY N. KOCHUBEI ◽  
MUSTAFA R. SAIT-AMETOV
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Differential Operator ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Quantum Field ◽  
Pseudo Differential Operator ◽  
Schwartz Distributions ◽  
Space Of Distributions ◽  
Euclidean Quantum Field Theory

We construct measures on the space [Formula: see text], n≤4, of Bruhat–Schwartz distributions over the field of p-adic numbers, corresponding to finite volume polynomial interactions in a p-adic analog of the Euclidean quantum field theory. In contrast to earlier results in this direction, our choice of the free measure is the Gaussian measure corresponding to an elliptic pseudo-differential operator over [Formula: see text]. Analogs of the Euclidean P(φ)-theories with free and half-Dirichlet boundary conditions are considered.

scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

scholarly journals On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Jaeyoung Chung ◽  
Jeongwook Chang
Keyword(s):  
Functional Equations ◽  
Generalized Functions ◽  
Ulam Stability ◽  
Schwartz Distributions ◽  
The Stability

We consider the Hyers-Ulam stability for a class of trigonometric functional equations in the spaces of generalized functions such as Schwartz distributions and Gelfand hyperfunctions.

LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES

2002 ◽  
Vol 05 (04) ◽  
pp. 483-502 ◽  
Author(s):  
A. YU. KHRENNIKOV ◽  
V. M. SHELKOVICH ◽  
O. G. SMOLYANOV
Keyword(s):  
Linear Space ◽  
Commutative Algebra ◽  
Linear Span ◽  
Generalized Functions ◽  
Colombeau Algebra ◽  
Functional Interpretation ◽  
Infinite Dimensional ◽  
Space Of Distributions ◽  
Vector Valued ◽  
Locally Convex

We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(fn(x)) finite from the left (i.e. fn(x)=0 for n<n0(f*)) whose components fn(x) belong to the linear span [Formula: see text] generated by the distributions δ(m-1)(x-ck), P((x-ck)-m), xm-1, where m=1, 2, …, ck ∈ ℝ, k = 1, …, s. The space of distributions [Formula: see text] can be realized as a subspace in [Formula: see text] This linear space [Formula: see text] has the structure of an associative and commutative algebra containing a unity element and free of zero divizors. The Schwartz counterexample does not hold in the algebra [Formula: see text]. Unlike the Colombeau algebra, whose elements have no explicit functional interpretation, elements of the algebra [Formula: see text] are infinite-dimensional Schwartz vector-valued distributions. This construction can be considered as a next step and a "model" on the way of constructing a nonlinear theory of distributions similar to that developed by L. Schwartz. The obtained results can be considerably generalized.

scholarly journals Generalized functions for applications

1985 ◽  
Vol 26 (3) ◽  
pp. 362-374 ◽  
Author(s):  
B. D. Craven
Keyword(s):  
Real Line ◽  
Generalized Functions ◽  
Generalized Function ◽  
The Real ◽  
Square Roots ◽  
Rigorous Approach ◽  
Schwartz Distributions ◽  
Delta Functions

AbstractA simple rigorous approach is given to generalized functions, suitable for applications. Here, a generalized function is defined as a genuine function on a superset of the real line, so that multiplication is unrestricted and associative, and various manipulations retain their classical meanings. The superset is simply constructed, and does not require Robinson's nonstandard real line. The generalized functions go beyond the Schwartz distributions, enabling products and square roots of delta functions to be discussed.

scholarly journals Eigenfunction expansion of generalized functions

1978 ◽  
Vol 72 ◽  
pp. 1-25 ◽  
Author(s):  
J. N. Pandey ◽  
R. S. Pathak
Keyword(s):  
Hilbert Space ◽  
Eigenfunction Expansion ◽  
Orthogonal Series ◽  
Integral Transforms ◽  
Generalized Functions ◽  
Series Expansions ◽  
Schwartz Distributions ◽  
Space Technique

Expansions of generalized functions have been investigated by many authors. Korevaar [11], Widlund [20], Giertz [8], Walter [19] developed procedures for expanding generalized functions of Korevaar [12], Temple [17], and Lighthill [13], Expansions of certain Schwartz distributions [15] into series of orthonormal functions were given by Zemanian [23] (see also Zemanian [24]) and thereby he extended a number of integral transforms to distributions. The method involved in his work is very much related to the Hilbert space technique and is of somewhat different character from those used in most of the works on integral transforms such as [24, chapters 1-8]. Other works that discuss orthogonal series expansions involving generalized functions are by Bouix [1, chapter 7], Braga and Schönberg [2], Gelfand and Shilov [7, vol. 3, chapter 4] and Warmbrod [21].

scholarly journals OPTIMAL EMBEDDINGS OF DISTRIBUTIONS INTO ALGEBRAS

2003 ◽  
Vol 46 (2) ◽  
pp. 373-378 ◽  
Author(s):  
H. Vernaeve
Keyword(s):  
Open Subset ◽  
Differential Algebra ◽  
Subject Classification ◽  
Partial Derivatives ◽  
Space Of Distributions ◽  
Primary 46F30 ◽  
Secondary 13C11

AbstractLet $\varOmega$ be a convex, open subset of $\mathbb{R}^n$ and let $\mathcal{D}'(\varOmega)$ be the space of distributions on $\varOmega$. It is shown that there exist linear embeddings of $\mathcal{D}'(\varOmega)$ into a differential algebra that commute with partial derivatives and that embed $\mathcal{C}^{\infty}(\varOmega)$ as a subalgebra. This embedding appears to be the first one after Colombeau’s to possess these properties. We show that many nonlinear operations on distributions can be defined that are not definable in the Colombeau setting.AMS 2000 Mathematics subject classification: Primary 46F30. Secondary 13C11

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