| 涂天亮.导数在边界上具有Lipα条件的调和函数插值逼近阶[J].数学年刊A辑,2015,36(1):1~12 |
| 导数在边界上具有Lipα条件的调和函数插值逼近阶 |
| Order of Interpolatory Approximation to Harmonic Functions with Lip α on the Boundary |
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| DOI： |
| 中文关键词: 调和函数, 调和插值多项式, 一致逼近, 存在性, 唯一性, 稳定性 |
| 英文关键词:Harmonic functions, Harmonic interpolatory
polynomials, Uniform approximation, Existence, Uniqueness, Stability |
| 基金项目:本文受到河南省自然科学基金 (No.20001110001) 的资助. |
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| 中文摘要: |
| D是由复平面z中一条Jordan闭曲线$\Ga$围成的单连区域, $z=0\in D$.函数$u(z)$在$D$内调和
且在$\Ga$上$u^{(q)}\in {\rm Lip}\al\ (0<\al<1)$.
基于复插值逼近理论证明了: 存在唯一的调和插值多项式
$u_n^*(z)$, 它与调和函数$u(z)$在$\Ga$的摄动Fej\'{e}r点$\{z_k^*\}_0^{n-1}$上有相同的值, 在$\o D$上一致收敛于$u(z)$,
且收敛是稳定的. 所得结果改进并推广了同类课题中已有的工作. |
| 英文摘要: |
| Let $D$ be a simply connected domain in the
complex z-plane bounded by a closed Jordan curve $\Ga$, $z=0\in D$,
and let the function $u(z)$ be harmonic in $D$ with $u^{(q)}\in {\rm
Lip}\al\ (0<\al<1)$ on $\Ga$. Based on the theory of complex approximation by
interpolation, it is proved that there exists a unique harmonic
interpolation polynomial $u_n^*(z)$ which coincides with $u(z)$ at
disturbed Fej\'{e}r points $\{z_k^*\}_0^{n-1}$ on $\Ga$, and uniformly
approximates to $u(z)$ on $\o D$. The convergence is stable. The
results obtained have improved or extended earlier similar works on
this topic. |
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