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

On the Decomposition of A Class of Functions of Bounded Variation

1964 ◽  
Vol 16 ◽  
pp. 479-484 ◽  
Author(s):  
R. G. Laha
Keyword(s):  
Distribution Function ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Distribution Functions ◽  
Characteristic Functions ◽  
Functions Of Bounded Variation ◽  
Image Position ◽  
Class Of Functions

Let F1(x) and F2(x) be two distribution functions, that is, non-decreasing, right-continuous functions such that Fj(— ∞) = 0 and Fj(+ ∞) = 1 (j = 1, 2). We denote their convolution by F(x) so thatthe above integrals being defined as the Lebesgue-Stieltjes integrals. Then it is easy to verify (2, p. 189) that F(x) is a distribution function. Let f1(t), f2(t), and f(t) be the corresponding characteristic functions, that is,

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

scholarly journals Perturbed Companions of Ostrowski’s Inequality for Absolutely Continuous Functions (I)

2016 ◽  
Vol 54 (1) ◽  
pp. 119-138
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Ostrowski’S Inequality

Abstract Perturbed companions of Ostrowski’s inequality for absolutely continuous functions whose derivatives are either bounded or of bounded variation and applications are given.

Absolute Continuity of Some Vector Functions and Measures

1972 ◽  
Vol 24 (5) ◽  
pp. 737-746 ◽  
Author(s):  
William J. Knight
Keyword(s):  
Bounded Variation ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Continuous Point ◽  
Uniform Boundedness Principle ◽  
Vector Valued Functions ◽  
Analogous Theorem ◽  
Vector Valued

In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.

scholarly journals Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation

2005 ◽  
Vol 2005 (23) ◽  
pp. 3827-3833 ◽  
Author(s):  
Vijay Gupta ◽  
Ulrich Abel ◽  
Mircea Ivan
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Continuous Functions ◽  
Function Of Bounded Variation ◽  
Approximation Properties ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Derivatives Of

We study the approximation properties of beta operators of second kind. We obtain the rate of convergence of these operators for absolutely continuous functions having a derivative equivalent to a function of bounded variation.

scholarly journals A note on convolution operators on Riesz Bounded variation spaces

2021 ◽  
Vol 40 (6) ◽  
pp. 1603-1613
Author(s):  
Lucía Guiterrez ◽  
Oscar M. Guzmán
Keyword(s):  
Bounded Variation ◽  
Continuous Functions ◽  
Convolution Type ◽  
Convolution Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

We show some estimates and approximation results of operators of convolution type defined on Riesz Bounded variation spaces in Rn. We also state some embedding results that involve the collection of generalized absolutely continuous functions.

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.

On Finite Part Integrals and Hadamard-Type Fractional Derivatives

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Li Ma ◽  
Changpin Li
Keyword(s):  
Fractional Derivative ◽  
Singular Integral ◽  
Fractional Derivatives ◽  
Continuous Functions ◽  
Finite Part ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Fundamental Properties ◽  
Logarithmic Series ◽  
The Right

This paper is devoted to investigating the relation between Hadamard-type fractional derivatives and finite part integrals in Hadamard sense; that is to say, the Hadamard-type fractional derivative of a given function can be expressed by the finite part integral of a strongly singular integral, which actually does not exist. Besides, our results also cover some fundamental properties on absolutely continuous functions, and the logarithmic series expansion formulas at the right end point of interval for functions in certain absolutely continuous spaces.

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