Uniformly continuous superposition operators in the space of bounded variation functions

2010 ◽  
Vol 283 (7) ◽  
pp. 1060-1064 ◽  
Author(s):  
Janusz Matkowski
Keyword(s):  
Bounded Variation ◽  
Superposition Operators ◽  
Bounded Variation Functions ◽  
Uniformly Continuous

Uniformly continuous composition operators in the space of bounded -variation functions

2010 ◽  
Vol 72 (6) ◽  
pp. 3119-3123 ◽  
Author(s):  
J.A. Guerrero ◽  
H. Leiva ◽  
J. Matkowski ◽  
N. Merentes
Keyword(s):  
Bounded Variation ◽  
Composition Operators ◽  
Bounded Variation Functions ◽  
Uniformly Continuous

scholarly journals A note on integral modification of the Meyer-König and Zeller operators

2003 ◽  
Vol 2003 (31) ◽  
pp. 2003-2009 ◽  
Author(s):  
Vijay Gupta ◽  
Niraj Kumar
Keyword(s):  
Rate Of Convergence ◽  
Bounded Variation ◽  
Present Note ◽  
Sharp Estimate ◽  
Functions Of Bounded Variation ◽  
Exact Bound ◽  
Bounded Variation Functions ◽  
Correct Estimate

Guo (1988) introduced the integral modification of Meyer-Kö nig and Zeller operatorsMˆnand studied the rate of convergence for functions of bounded variation. Gupta (1995) gave the sharp estimate for the operatorsMˆn. Zeng (1998) gave the exact bound and claimed to improve the results of Guo and Gupta, but there is a major mistake in the paper of Zeng. In the present note, we give the correct estimate for the rate of convergence on bounded variation functions.

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.

scholarly journals On the Nemytskii Operator in the Space of Functions of Bounded (p, 2, α)-Variation with Respect to the Weight Function

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Wadie Aziz
Keyword(s):  
Weight Function ◽  
Bounded Variation ◽  
The Other ◽  
Affine Function ◽  
Nemytskii Operator ◽  
Other Hand ◽  
Generator Function ◽  
Uniformly Continuous

AbstractIn this paper, we consider the Nemytskii operator (Hf)(t) = h(t, f(t)), generated by a given function h. It is shown that if H is globally Lipschitzian and maps the space of functions of bounded (p,2,α)-variation (with respect to a weight function α) into the space of functions of bounded (q,2,α)-variation (with respect to α) 1<q<p, then H is of the form (Hf)(t) = A(t)f(t)+B(t). On the other hand, if 1<p<q then H is constant. It generalize several earlier results of this type due to Matkowski-Merentes and Merentes. Also, we will prove that if a uniformly continuous Nemytskii operator maps a space of bounded variation with weight function in the sense of Merentes into another space of the same type, its generator function is an affine function.

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