<p>Neural network (NN) operators are widely recognized for providing a constructive approach to approximate given class of functions. In this paper, we study the convergence of a family of semi-discrete NN operators, known as <i>Durrmeyer</i> type NN operators, in the general framework of <i>Orlicz spaces</i> on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_685_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">\([a,b] (\subset \mathbb {R}),\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mo>⊂</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> denoted by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_685_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\phi }([a,b]).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>ϕ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The Orlicz space consists of various function spaces, including classical <i>Lebesgue spaces, Exponential spaces, Zygmund spaces</i>, for different choices of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_685_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\phi -\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>-</mo> </mrow> </math></EquationSource> </InlineEquation>functions. Hence this note presents a unified approximation procedure for these operators across various function spaces. We establish the boundedness of Durrmeyer type NN operators within <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11868_2025_685_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="70" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\phi }([a,b]).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>ϕ</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Further, modular convergence theorem is deduced for these NN operators in general setting of Orlicz spaces. Lastly, we enclose some graphical representations and error-estimates to demonstrate the approximation process.</p>

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Modular convergence of semi-discrete neural network operators in Orlicz space

  • Priyanka Majethiya,
  • Shivam Bajpeyi

摘要

Neural network (NN) operators are widely recognized for providing a constructive approach to approximate given class of functions. In this paper, we study the convergence of a family of semi-discrete NN operators, known as Durrmeyer type NN operators, in the general framework of Orlicz spaces on \([a,b] (\subset \mathbb {R}),\) [ a , b ] ( R ) , denoted by \(L^{\phi }([a,b]).\) L ϕ ( [ a , b ] ) . The Orlicz space consists of various function spaces, including classical Lebesgue spaces, Exponential spaces, Zygmund spaces, for different choices of \(\phi -\) ϕ - functions. Hence this note presents a unified approximation procedure for these operators across various function spaces. We establish the boundedness of Durrmeyer type NN operators within \(L^{\phi }([a,b]).\) L ϕ ( [ a , b ] ) . Further, modular convergence theorem is deduced for these NN operators in general setting of Orlicz spaces. Lastly, we enclose some graphical representations and error-estimates to demonstrate the approximation process.