A Nuclear Space of Distributions on a Space of Incomplete Continuous Functions

Let H be a real separable Hilbert space and let E⊂H be a nuclear space with the chain {Em: m=1,2,…} of Hilbert spaces such that E = ∩m∈ℕEm. Let E* and E-m denote the dual spaces of E and Em, respectively. For γ > 0, let [Formula: see text] be the collection of complex-valued continuous functions f defined on E* such that [Formula: see text] is finite for every m. Then [Formula: see text] is a complete countably normed space equipping with the family {‖·‖m,γ : m = 1,2,…} of norms. Using a probabilistic approach, it is shown that every continuous linear functional T on [Formula: see text] can be represented uniquely by a complex Borel measure νT satisfying certain exponential integrability condition. The results generalize an infinite dimensional Riesz representation theorem given previously by the first author for the case γ = 2. As applications, we establish a Weierstrass approximation theorem on E* for γ≥1 and show that the space [Formula: see text] spanned by the class { exp [i(x,ξ)] : ξ ∈ E} is dense in [Formula: see text] for γ>0. In the course of the proof we give sufficient conditions for a function space on which every positive functional can be represented by a Borel measure on E*.

Download Full-text

Approximation in statistical sense to B -continuous functions by positive linear operators

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2009.1129 ◽

2010 ◽

Vol 47 (3) ◽

pp. 289-298 ◽

Cited By ~ 2

Author(s):

Fadime Dirik ◽

Oktay Duman ◽

Kamil Demirci

Keyword(s):

Compact Subset ◽

Real Line ◽

Statistical Convergence ◽

Approximation Theorem ◽

Continuous Functions ◽

Linear Operators ◽

Positive Linear Operators ◽

Statistical Approximation ◽

Classical Version ◽

Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

Download Full-text

Almost Somewhat Near Continuity and Near Regularity

Moroccan Journal of Pure and Applied Analysis ◽

10.2478/mjpaa-2021-0009 ◽

2021 ◽

Vol 7 (1) ◽

pp. 88-99

Author(s):

Zanyar A. Ameen

Keyword(s):

Continuous Function ◽

Continuous Functions ◽

One To One ◽

Nearly Continuous

AbstractThe notions of almost somewhat near continuity of functions and near regularity of spaces are introduced. Some properties of almost somewhat nearly continuous functions and their connections are studied. At the end, it is shown that a one-to-one almost somewhat nearly continuous function f from a space X onto a space Y is somewhat nearly continuous if and only if the range of f is nearly regular.

Download Full-text

100-kWe nuclear space electric power source

10.2514/6.1979-2089 ◽

1979 ◽

Cited By ~ 5

Author(s):

D. BUDEN

Keyword(s):

Electric Power ◽

Power Source ◽

Nuclear Space

Download Full-text

Approximating Continuous Functions by Iterated Function Systems and Optimization

SSRN Electronic Journal ◽

10.2139/ssrn.616605 ◽

2002 ◽

Author(s):

Davide La Torre ◽

Matteo Rocca

Keyword(s):

Iterated Function Systems ◽

Continuous Functions ◽

Iterated Function

Download Full-text

On existence result of a class of nonlinear integral equation

Filomat ◽

10.2298/fil1711593b ◽

2017 ◽

Vol 31 (11) ◽

pp. 3593-3597

Author(s):

Ravindra Bisht

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Existence And Uniqueness ◽

Existence Result ◽

Nonlinear Integral Equation ◽

Continuous Functions ◽

Concave Functions ◽

Real Numbers ◽

Fixed Point Techniques ◽

Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Download Full-text