scholarly journals Characterizing the continuous functionals

1983 ◽  
Vol 48 (4) ◽  
pp. 965-969 ◽  
Author(s):  
Dag Normann
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Rational Numbers ◽  
The Real ◽  
Finite Systems ◽  
Several Variables ◽  
Finite Structures ◽  
Starting Point ◽  
Topological Completion ◽  
Continuous Functionals

One of the objectives of mathematics is to construct suitable models for practical or theoretical phenomena and to explore the mathematical richness of such models. This enables other scientists to obtain a better understanding of such phenomena. As an example we will mention the real line and related structures. The line can be used profitably in the study of discrete phenomena like population growth, chemical reactions, etc.Today's version of the real line is a topological completion of the rational numbers. This is so because then mathematicians have been able to work out a powerful analysis of the line. By using the real line to construct models for finitary phenomena we are more able to study those phenomena than we would have been sticking only to true-to-nature but finite structures.So we may say that the line is a mathematical model for certain finite structures. This motivates us to seek natural models for other types of finite structures, and it is natural to look for models that in some sense are complete.In this paper our starting point will be finite systems of finite operators. For the sake of simplicity we assume that they all are operators of one variable and that all the values are natural numbers. There is a natural extension of the systems such that they accept several variables and give finite operators as values, but the notational complexity will then obscure the idea of the construction.

scholarly journals On the Fourier transforms

1985 ◽  
Vol 31 (2) ◽  
pp. 171-179
Author(s):  
Hwai-chiuan Wang
Keyword(s):  
Fourier Transform ◽  
Convex Function ◽  
Real Line ◽  
Integrable Function ◽  
Fourier Transforms ◽  
The Real ◽  
Factorization Problem ◽  
Several Variables ◽  
The Fourier Transform

In this article we give a new proof of the theorem that a positive even convex function on the real line, which vanishes at infinity, is the Fourier transform of an integrable function. Related results in several variables are also proved. As an application of our results we solve the factorization problem of Sobolev algebras.

scholarly journals Generalized Exponential Trichotomies for Abstract Evolution Operators on the Real Line

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Nicolae Lupa ◽  
Mihail Megan
Keyword(s):  
Differential Equations ◽  
Real Line ◽  
Exponential Dichotomy ◽  
Natural Extension ◽  
Evolution Operators ◽  
The Real ◽  
Exponential Trichotomy ◽  
Whole Space

This paper considers two trichotomy concepts in the context of abstract evolution operators. The first one extends the notion of exponential trichotomy in the sense of Elaydi-Hajek for differential equations to abstract evolution operators, and it is a natural extension of the generalized exponential dichotomy considered in the paper by Jiang (2006). The second concept is dual in a certain sense to the first one. We prove that these types of exponential trichotomy imply the existence of generalized exponential dichotomy on both half-lines. We emphasize that we do not assume the invertibility of the evolution operators on the whole spaceX(unlike the case of evolution operators generated by differential equations).

scholarly journals On the Group of Homeomorphisms of the Real Line That Map the Pseudoboundary Onto Itself

2006 ◽  
Vol 58 (3) ◽  
pp. 529-547 ◽  
Author(s):  
Jan J. Dijkstra ◽  
Jan van Mill
Keyword(s):  
Real Line ◽  
Rational Numbers ◽  
Cantor Set ◽  
Vietoris Topology ◽  
Compact Open Topology ◽  
Product Topology ◽  
The Real ◽  
Group Of Homeomorphisms ◽  
Nonempty Compact ◽  
Infinite Power

AbstractIn this paper we primarily consider two natural subgroups of the autohomeomorphism group of the real line ℝ, endowed with the compact-open topology. First, we prove that the subgroup of homeomorphisms that map the set of rational numbers ℚ onto itself is homeomorphic to the infinite power of ℚ with the product topology. Secondly, the group consisting of homeomorphisms that map the pseudoboundary onto itself is shown to be homeomorphic to the hyperspace of nonempty compact subsets of ℚ with the Vietoris topology. We obtain similar results for the Cantor set but we also prove that these results do not extend to ℝn for n ≥ 2, by linking the groups in question with Erdős space.

scholarly journals The special atom space and Haar wavelets in higher dimensions

2020 ◽  
Vol 53 (1) ◽  
pp. 131-151
Author(s):  
Eddy Kwessi ◽  
Geraldo de Souza ◽  
Ngalla Djitte ◽  
Mariama Ndiaye
Keyword(s):  
Real Line ◽  
Natural Extension ◽  
Higher Dimensions ◽  
Haar Wavelets ◽  
Lipschitz Spaces ◽  
The Real ◽  
Haar Functions ◽  
Wide Range ◽  
Definition Of ◽  
Almost All

AbstractIn this note, we will revisit the special atom space introduced in the early 1980s by Geraldo De Souza and Richard O’Neil. In their introductory work and in later additions, the space was mostly studied on the real line. Interesting properties and connections to spaces such as Orlicz, Lipschitz, Lebesgue, and Lorentz spaces made these spaces ripe for exploration in higher dimensions. In this article, we extend this definition to the plane and space and show that almost all the interesting properties such as their Banach structure, Hölder’s-type inequalities, and duality are preserved. In particular, dual spaces of special atom spaces are natural extension of Lipschitz and generalized Lipschitz spaces of functions in higher dimensions. We make the point that this extension could allow for the study of a wide range of problems including a connection that leads to what seems to be a new definition of Haar functions, Haar wavelets, and wavelets on the plane and on the space.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

AN OSCILLATION FUNCTION ON THE REAL LINE

2000 ◽  
Vol 26 (1) ◽  
pp. 237
Author(s):  
Duszyński
Keyword(s):  
Real Line ◽  
The Real

DERIVATION BASES ON THE REAL LINE (I)

1982 ◽  
Vol 8 (1) ◽  
pp. 67 ◽  
Author(s):  
Thomson
Keyword(s):  
Real Line ◽  
The Real

On finite sums of periodic functions

2020 ◽  
Vol 27 (2) ◽  
pp. 265-269
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Analytic Function ◽  
Real Line ◽  
Real Analytic Function ◽  
Periodic Functions ◽  
Uniform Limit ◽  
The Real ◽  
Pointwise Limit ◽  
Finite Sums ◽  
Real Analytic

AbstractIt is shown that any function acting from the real line {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

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