<p>We characterize the family of continuous functions <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2348_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">$f\in C([0,1])$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that the iterates <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2348_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{T}_i^kf$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>i</mi> <mi>k</mi> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> converge uniformly on [0, 1], where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2024_2348_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{T}_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover accent="true"> <mi>T</mi> <mo stretchy="true">^</mo> </mover> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation> is a generalized Kantorovich operator. This gives an affirmative answer to the problem raised in 2021 by Acu and Rasa. </p>

Asymptotic Dynamics of Generalized Kantorovich Operators

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2025年1月18日

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Krzysztof Bartoszek,
Wojciech Bartoszek

摘要

We characterize the family of continuous functions $f\in C([0,1])$ $f \in C ([0, 1])$ such that the iterates $\widehat{T}_i^kf$ ${\hat{T}}_{i}^{k} f$ converge uniformly on [0, 1], where $\widehat{T}_i$ ${\hat{T}}_{i}$ is a generalized Kantorovich operator. This gives an affirmative answer to the problem raised in 2021 by Acu and Rasa.