scholarly journals Regular variation for measures on metric spaces

2006 ◽  
Vol 80 (94) ◽  
pp. 121-140 ◽  
Author(s):  
Henrik Hult ◽  
Filip Lindskog
Keyword(s):  
Regular Variation ◽  
Metric Spaces ◽  
General Setting ◽  
Continuous Functions ◽  
Separable Space ◽  
Real Numbers ◽  
Space Of Continuous Functions ◽  
Relative Compactness ◽  
Borel Measures

The foundations of regular variation for Borel measures on a complete separable space S, that is closed under multiplication by nonnegative real numbers, is reviewed. For such measures an appropriate notion of convergence is presented and the basic results such as a Portmanteau theorem, a mapping theorem and a characterization of relative compactness are derived. Regular variation is defined in this general setting and several statements that are equivalent to this definition are presented. This extends the notion of regular variation for Borel measures on the Euclidean space Rd to more general metric spaces. Some examples, including regular variation for Borel measures on Rd, the space of continuous functions C and the Skorohod space D, are provided.

Existence of unique solution to nonlinear mixed Volterra Fredholm-Hammerstein integral equations in complex-valued fuzzy metric spaces

2020 ◽  
pp. 1-10
Author(s):  
Humaira ◽  
Muhammad Sarwar ◽  
Thabet Abdeljawad
Keyword(s):  
Unique Solution ◽  
Metric Spaces ◽  
General Setting ◽  
Continuous Functions ◽  
Contractive Condition ◽  
Nonlinear Functions ◽  
Fuzzy Metric Spaces ◽  
Fuzzy Metric ◽  
Hammerstein Integral Equations ◽  
Complex Valued

The purpose of this article is to investigate the existence of unique solution for the following mixed nonlinear Volterra Fredholm-Hammerstein integral equation considered in complex plane; (0.1) ξ ( τ ) = g ( t ) + ρ ∫ 0 τ K 1 ( τ , ℘ ) ϝ 1 ( ℘ , ξ ( ℘ ) ) d ℘ + ϱ ∫ 0 1 K 2 ( τ , ℘ ) ϝ 2 ( ℘ , ξ ( ℘ ) ) d ℘ , such that ξ = ξ 1 + ξ 2 , ξ 1 , ξ 2 ∈ ( C ( [ 0 , 1 ] ) , R ) g = g 1 + g 2 , g l : [ 0 , 1 ] → R , l = 1 , 2 , ϝ l ( ℘ , ξ ( ℘ ) ) = ϝ l 1 * ( ℘ , ξ 1 * ) + i ϝ l 2 * ( ℘ , ξ 2 * ) , ϝ lj * : [ 0 , 1 ] × R → R for l , j = 1 , 2 , and ξ 1 * , ξ 2 * ∈ ( C ( [ 0 , 1 ] ) , R ) K l ( t , ℘ ) = K l 1 * ( t , ℘ ) + iK l 2 * ( t , ℘ ) , for l , j = 1 , 2 and K lj * : [ 0 , 1 ] 2 → R , where ρ and ϱ are constants, g (t), the kernels K l  (τ, ℘) and the nonlinear functions ϝ1 (℘ , ξ (℘)) , ϝ 2 (℘ , ξ (℘)) are continuous functions on the interval 0 ≤ τ ≤ 1. In this direction we apply fixed point results for self mappings with the concept of (ψ, ϕ) contractive condition in the setting of complex-valued fuzzy metric spaces. This study will be useful in the development of the theory of fuzzy fractional differential equations in a more general setting.

Free Vector Lattices

1968 ◽  
Vol 20 ◽  
pp. 58-66 ◽  
Author(s):  
Kirby A. Baker
Keyword(s):  
Universal Algebra ◽  
Vector Lattice ◽  
Piecewise Linear ◽  
Continuous Functions ◽  
Real Numbers ◽  
Vector Lattices ◽  
Explicit Characterization

This note presents a useful explicit characterization of the free vector lattice FVL(ℵ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL(ℵ) has been given by Weinberg (14) and simplified by Holland (13, § 5). Weinberg's construction yields the fact that FVL(ℵ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.

scholarly journals A Helly theorem for functions with values in metric spaces

2009 ◽  
Vol 44 (1) ◽  
pp. 159-168 ◽  
Author(s):  
Miloslav Duchoň ◽  
Peter Maličký
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Type Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Space Of Continuous Functions ◽  
Riesz Theorem ◽  
Helly Theorem

Abstract We present a Helly type theorem for sequences of functions with values in metric spaces and apply it to representations of some mappings on the space of continuous functions. A generalization of the Riesz theorem is formulated and proved. More concretely, a representation of certain majored linear operators on the space of continuous functions into a complete metric space.

scholarly journals Multipliers from spaces of test functions to amalgams

1993 ◽  
Vol 54 (1) ◽  
pp. 97-110
Author(s):  
Maria Torres De Squire
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Continuous Functions ◽  
Locally Compact Abelian Group ◽  
Test Functions ◽  
Space Of Continuous Functions ◽  
Locally Compact ◽  
The Fourier Transform

AbstractIn this paper we study the space of multipliers M(r, s: p, q) from the space of test functions Φrs(G), on a locally compact abelian group G, to amalgams (Lp, lq)(G); the former includes (when r = s = ∞) the space of continuous functions with compact support and the latter are extensions of the Lp(G) spaces. We prove that the space M(∞: p) is equal to the derived space (Lp)0 defined by Figá-Talamanca and give a characterization of the Fourier transform for amalgams in terms of these spaces of multipliers.

scholarly journals Characterization of pseudocompactness by the topology of uniform convergence on function spaces

1978 ◽  
Vol 26 (2) ◽  
pp. 251-256 ◽  
Author(s):  
R. A. McCoy
Keyword(s):  
Uniform Convergence ◽  
Continuous Functions ◽  
Function Spaces ◽  
Metrizable Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Space Of Continuous Functions ◽  
Tychonoff Space ◽  
Necessary And Sufficient

AbstractIt is shown that a Tychonoff space X is pseudocompact if and only if for every metrizable space Y, all uniformities on Y induce the same topology on the space of continuous functions from X into Y. Also for certain pairs of spaces X and Y, a necessary and sufficient condition is established in order that all uniformities on Y induce the same topology on the space of continuous functions from X into Y.

scholarly journals A monadic, functional implementation of real numbers

2007 ◽  
Vol 17 (1) ◽  
pp. 129-159 ◽  
Author(s):  
RUSSELL ÓCONNOR
Keyword(s):  
Real Number ◽  
Large Scale ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Proof Assistants ◽  
Elementary Functions ◽  
Real Numbers ◽  
Mathematical Proofs ◽  
Regular Functions ◽  
Regular Sequences

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion (Bishop and Bridges 1985) of regular sequences is generalised to what I call regular functions, which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (which are a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to verify its correctness easily.

Some properties of quasisymmetries in metric spaces

2019 ◽  
Vol 16 (1) ◽  
pp. 2-9
Author(s):  
Elena Afanas'eva ◽  
Victoria Bilet
Keyword(s):  
Metric Spaces ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Locally Finite ◽  
Borel Measures ◽  
Compact Subsets

Let \((X, d, \mu)\) and \((Y, d', \mu')\) be metric spaces \(\alpha\)-regular by Ahlfors with \(\alpha>0\) and locally finite Borel measures \(\mu\) and \(\mu'\), respectively. We consider the class \(ACS_{E}\) of absolutely continuous functions on a.a. compact subsets \(E\subset X\) and establish the membership of mappings \(f: X\to Y\) to a given class.

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