<p>The present paper provides the study on the composition operators due to Rathore and Charlier based operators. We consider the composition of the operators based on Charlier polynomials with Rathore operators on both way i.e. <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1737_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$M_m^{a}\circ W_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>M</mi> <mi>m</mi> <mi>a</mi> </msubsup> <mo>∘</mo> <msub> <mi>W</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1737_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="69" /> </InlineMediaObject> <EquationSource Format="TEX">$W_n \circ M_m^{a}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>n</mi> </msub> <mo>∘</mo> <msubsup> <mi>M</mi> <mi>m</mi> <mi>a</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation>. Also, we establish difference of such operators with original ones, error estimation for exponential functions and quantitative asymptotic estimate. We conclude that in limiting case both compositions give same results but the quantitative estimate shows that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1737_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$W_n \circ M_n^{a}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>n</mi> </msub> <mo>∘</mo> <msubsup> <mi>M</mi> <mi>n</mi> <mi>a</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> gives better approximation in comparison to <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2025_1737_Article_IEq4.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$M_n^{a}\circ W_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>M</mi> <mi>n</mi> <mi>a</mi> </msubsup> <mo>∘</mo> <msub> <mi>W</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. In the end, graphical comparison is indicated.</p>

Convergence of Charlier type operators and composition

期刊论文
发表日期：

2025年5月22日

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Vijay Gupta

摘要

The present paper provides the study on the composition operators due to Rathore and Charlier based operators. We consider the composition of the operators based on Charlier polynomials with Rathore operators on both way i.e. $M_m^{a}\circ W_n$ $M_{m}^{a} \circ W_{n}$ and $W_n \circ M_m^{a}$ $W_{n} \circ M_{m}^{a}$ . Also, we establish difference of such operators with original ones, error estimation for exponential functions and quantitative asymptotic estimate. We conclude that in limiting case both compositions give same results but the quantitative estimate shows that $W_n \circ M_n^{a}$ $W_{n} \circ M_{n}^{a}$ gives better approximation in comparison to $M_n^{a}\circ W_n$ $M_{n}^{a} \circ W_{n}$ . In the end, graphical comparison is indicated.