Methods of comparison of families of real functions in porosity terms

2019 ◽  
Vol 26 (4) ◽  
pp. 643-654 ◽  
Author(s):  
Stanisław Kowalczyk ◽  
Małgorzata Turowska
Keyword(s):  
Uniform Convergence ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Absolutely Continuous ◽  
Real Functions ◽  
Upper Continuous ◽  
Baire One ◽  
Lower Continuous

Abstract We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.

A Distribution Function of Cantor-Vitali Type

1964 ◽  
Vol 7 (1) ◽  
pp. 65-75
Author(s):  
D. B. Sumner
Keyword(s):  
Distribution Function ◽  
Characteristic Function ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Characteristic Functions ◽  
Functions Of Bounded Variation ◽  
Absolutely Continuous Functions ◽  
Full Account ◽  
Absolutely Continuous ◽  
Bernoulli Distributions

In his 1922 article [l] on functions of bounded variation, Vitali gave a method for constructing monotone non-absolutely continuous functions, generalizing ideas from the ternary set introduced in another connection by Cantor. In [2], Hille and Tamarkin gave a full account of the "middle-third" function, showing it to be a singular distribution function, and finding its characteristic function. In [3], Evans obtained a generalization of the middle - third function by discarding middle intervals of length other than one-third, and obtained algorithms by which the moments of his function could be calculated. Invarious papers, among them [4], Wintner studied infinite convolutions of symmetric Bernoulli distributions, finding a great variety of distributions whose characteristic functions were of the form

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

scholarly journals The Nemitskij operator on Lipk-type and BVk-type spaces

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
José Giménez ◽  
Lorena López ◽  
N. Merentes
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Function Spaces ◽  
Compact Interval ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Type Spaces

AbstractIn this paper, we discuss and present various results about acting and boundedness conditions of the autonomous Nemitskij operator on certain function spaces related to the space of all real valued Lipschitz (of bounded variation, absolutely continuous) functions defined on a compact interval of ℝ. We obtain a result concerning the integrability of products of the form

DEFINITION AND CLASSIFICATION OF ONE-DIMENSIONAL CONTINUOUS FUNCTIONS WITH UNBOUNDED VARIATION

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750048 ◽  
Author(s):  
Y. S. LIANG
Keyword(s):  
Fractal Dimension ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Nondifferentiable Functions ◽  
Differentiable Functions ◽  
Closed Intervals ◽  
Bounded Variation Functions

The present paper mainly investigates the definition and classification of one-dimensional continuous functions on closed intervals. Continuous functions can be classified as differentiable functions and nondifferentiable functions. All differentiable functions are of bounded variation. Nondifferentiable functions are composed of bounded variation functions and unbounded variation functions. Fractal dimension of all bounded variation continuous functions is 1. One-dimensional unbounded variation continuous functions may have finite unbounded variation points or infinite unbounded variation points. Number of unbounded variation points of one-dimensional unbounded variation continuous functions maybe infinite and countable or uncountable. Certain examples of different one-dimensional continuous functions have been given in this paper. Thus, one-dimensional continuous functions are composed of differentiable functions, nondifferentiable continuous functions of bounded variation, continuous functions with finite unbounded variation points, continuous functions with infinite but countable unbounded variation points and continuous functions with uncountable unbounded variation points. In the end of the paper, we give an example of one-dimensional continuous function which is of unbounded variation everywhere.

Approximation of functions of bounded variation by integrated Meyer-König and Zeller operators of finite type

2012 ◽  
Vol 49 (2) ◽  
pp. 254-268
Author(s):  
Tiberiu Trif
Keyword(s):  
Rate Of Convergence ◽  
Finite Type ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Approximation Of Functions ◽  
Durrmeyer Operators

I. Gavrea and T. Trif [Rend. Circ. Mat. Palermo (2) Suppl. 76 (2005), 375–394] introduced a class of Meyer-König-Zeller-Durrmeyer operators “of finite type” and investigated the rate of convergence of these operators for continuous functions. In the present paper we study the approximation of functions of bounded variation by means of these operators.

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