scholarly journals The Composition Operator and the Space of the Functions of Bounded Variation in Schramm-Korenblum's Sense

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Z. Jesús ◽  
O. Mejía ◽  
N. Merentes ◽  
S. Rivas
Keyword(s):  
Bounded Variation ◽  
Composition Operator ◽  
Functions Of Bounded Variation ◽  
Locally Lipschitz ◽  
Uniformly Bounded

We show that the composition operatorH, associated withh:[a,b]→ℝ, maps the spacesLip[a,b]on to the spaceκBVϕa,bof functions of bounded variation in Schramm-Korenblum's sense if and only ifhis locally Lipschitz. Also, verify that if the composition operator generated byh:[a,b]×ℝ→ℝmaps this space into itself and is uniformly bounded, then regularization ofhis affine in the second variable.

The form of locally defined operators in waterman spaces

2021 ◽  
Vol 71 (6) ◽  
pp. 1529-1544
Author(s):  
Małgorzata Wróbel
Keyword(s):  
Banach Spaces ◽  
Bounded Variation ◽  
Composition Operators ◽  
Representation Formula ◽  
Continuous Functions ◽  
Functions Of Bounded Variation ◽  
Spaces Of Continuous Functions ◽  
Local Operators ◽  
Uniformly Bounded

Abstract A representation formula for locally defined operators acting between Banach spaces of continuous functions of bounded variation in the Waterman sense is presented. Moreover, the Nemytskij composition operators will be investigated and some consequences for locally bounded as well as uniformly bounded local operators will be given.

scholarly journals Uniformly bounded set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener

2015 ◽  
Vol 14 (4) ◽  
pp. 41-51
Author(s):  
José Atilio Guerrero ◽  
◽  
Janusz Matkowski ◽  
Nelson Merentes ◽  
Małgorzata Wróbel ◽  
...  
Keyword(s):  
Bounded Variation ◽  
Composition Operators ◽  
Functions Of Bounded Variation ◽  
Bounded Set ◽  
Uniformly Bounded

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 Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 990
Author(s):  
Silvestru Sever Dragomir
Keyword(s):  
Bounded Variation ◽  
Functions Of Bounded Variation ◽  
Functional Applications ◽  
Čebyšev Functional

In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided.

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.

On the regularity of one-sided fractional maximal functions

2018 ◽  
Vol 68 (5) ◽  
pp. 1097-1112 ◽  
Author(s):  
Feng Liu
Keyword(s):  
Bounded Variation ◽  
Maximal Functions ◽  
Continuous Case ◽  
Discrete Case ◽  
Maximal Operators ◽  
Functions Of Bounded Variation ◽  
Sharp Bounds ◽  
Regularity Properties ◽  
Natural Extensions ◽  
The One

Abstract In this paper we investigate the regularity properties of one-sided fractional maximal functions, both in continuous case and in discrete case. We prove that the one-sided fractional maximal operators $ \mathcal{M}_{\beta}^{+} $ and $ \mathcal{M}_{\beta}^{-} $ map $ W^{1,p}(\mathbb{R}) $ into $ W^{1,q}(\mathbb{R}) $ with 1 <p <∞, 0≤β<1/p and q=p/(1-pβ), boundedly and continuously. In addition, we also obtain the sharp bounds and continuity for the discrete one-sided fractional maximal operators $ M_{\beta}^{+} $ and $ M_{\beta}^{-} $ from $ \ell^{1}(\mathbb{Z}) $ to $ {\rm BV}(\mathbb{Z}) $. Here $ {\rm BV}(\mathbb{Z}) $ denotes the set of all functions of bounded variation defined on ℤ. The results we obtained represent significant and natural extensions of what was known previously.

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