scholarly journals Bounded variation functions and some properties on metric spaces

2021 ◽  
Vol 1943 (1) ◽  
pp. 012126
Author(s):  
I D Murdianingsih ◽  
S Hariyanto ◽  
T Udjiani ◽  
Y D Sumanto
Keyword(s):  
Bounded Variation ◽  
Metric Spaces ◽  
Bounded Variation Functions

scholarly journals Metric Fourier Approximation of Set-Valued Functions of Bounded Variation

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Elena E. Berdysheva ◽  
Nira Dyn ◽  
Elza Farkhi ◽  
Alona Mokhov
Keyword(s):  
Fourier Series ◽  
Bounded Variation ◽  
Error Bounds ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Dirichlet Kernel ◽  
Partial Sums ◽  
Functions Of Bounded Variation ◽  
Moduli Of Continuity ◽  
Fourier Approximation

AbstractWe introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial sums converges pointwisely in the Hausdorff metric to the values of the approximated set-valued function at its points of continuity, or to a certain set described in terms of the metric selections of the approximated multifunction at a point of discontinuity. Our error bounds are obtained with the help of the new notions of one-sided local moduli and quasi-moduli of continuity which we discuss more generally for functions with values in metric spaces.

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 The Jordan decomposition of bounded variation functions valued in vector spaces

2017 ◽  
Vol 2 (4) ◽  
pp. 635-646
Author(s):  
Francisco J. Mendoza-Torres ◽  
◽  
Juan A. Escamilla-Reyna ◽  
Daniela Rodríguez-Tzompantzi
Keyword(s):  
Bounded Variation ◽  
Vector Spaces ◽  
Jordan Decomposition ◽  
Bounded Variation Functions

Pointwise properties of functions of bounded variation in metric spaces

2013 ◽  
Vol 27 (1) ◽  
pp. 41-67 ◽  
Author(s):  
Juha Kinnunen ◽  
Riikka Korte ◽  
Nageswari Shanmugalingam ◽  
Heli Tuominen
Keyword(s):  
Bounded Variation ◽  
Metric Spaces ◽  
Functions Of Bounded Variation
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