<p>In this paper we prove asymptotic formulae for the sequences of linear positive operators on the space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1735_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$C_{2\pi }\left( \left[ \alpha ,\beta \right] \times \mathbb {R}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </msub> <mfenced close=")" open="("> <mfenced close="]" open="["> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mfenced> <mo>×</mo> <mi mathvariant="double-struck">R</mi> </mfenced> </mrow> </math></EquationSource> </InlineEquation> of the all bivariate continuous functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1735_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="133" /> </InlineMediaObject> <EquationSource Format="TEX">$f: \left[ \alpha ,\beta \right] \times \mathbb {R}\rightarrow \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mfenced close="]" open="["> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mfenced> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation> which are <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1735_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$2\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation>-periodic with respect to the second variable. Some examples are given.</p>

Asymptotic formulae for bivariate functions periodic with respect to the one variable

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2025年5月7日

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Dumitru Popa

摘要

In this paper we prove asymptotic formulae for the sequences of linear positive operators on the space $C_{2\pi }\left( \left[ \alpha ,\beta \right] \times \mathbb {R}\right) $ $C_{2 π} ([α, β] \times R)$ of the all bivariate continuous functions $f: \left[ \alpha ,\beta \right] \times \mathbb {R}\rightarrow \mathbb {R}$ $f : [α, β] \times R \to R$ which are $2\pi $ $2 π$ -periodic with respect to the second variable. Some examples are given.