<p>In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_n^{j,i}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>n</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i,j\in {\mathbb {N}}_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <msub> <mi mathvariant="double-struck">N</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$i=j=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, these operators reduce to the genuine Bernstein–Durrmeyer operators, while for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$i=j=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>=</mo> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_n^{j,i}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>L</mi> <mi>n</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msubsup> </math></EquationSource> </InlineEquation> under suitable smoothness conditions on the approximated function.</p>

Asymptotic Expansions for Generalized Bernstein–Durrmeyer and Genuine Bernstein–Durrmeyer Operators

期刊论文
出版模式：

开放获取
发表日期：

2026年3月4日

查看全文

Ulrich Abel,
Ana Maria Acu,
Margareta Heilmann,
Ioan Raşa

摘要

In this paper, we study an asymptotic expansion for a general family of Bernstein–Durrmeyer type operators $L_n^{j,i}$ $L_{n}^{j, i}$ , $i,j\in {\mathbb {N}}_0$ $i, j \in N_{0}$ . For $i=j=1$ $i = j = 1$ , these operators reduce to the genuine Bernstein–Durrmeyer operators, while for $i=j=0$ $i = j = 0$ they coincide with the classical Bernstein–Durrmeyer operators. We derive an explicit asymptotic expansion for $L_n^{j,i}$ $L_{n}^{j, i}$ under suitable smoothness conditions on the approximated function.