发表机构
Imam Mohammad Ibn Saud Islamic University (IMSIU)（伊玛目穆罕默德·本·沙特伊斯兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出后向对数正交函数及其广义形式，用于谱逼近终端端点奇异问题，发展了完整逼近理论，数值实验表明可恢复指数或高阶收敛速度。
AI 中文摘要
我们引入了一类新的后向对数正交函数和广义后向对数正交函数，它们是通过对广义拉盖尔多项式应用终端端点对数映射而构造的。这些函数旨在用于对解在终端端点处表现出弱奇异性的问题进行后向谱逼近。所提出的基函数生成非多项式加权逼近空间，其节点自然聚集在奇异端点附近，因此为解决代数和对数端点奇异性提供了一个有效的框架。我们发展了这些后向对数正交函数的基本逼近理论，包括递推关系、导数公式、正交性、Sturm-Liouville刻画、映射的Laguerre-Gauss求积规则、加权投影估计、后向Lagrange插值估计、逆不等式，以及通过终端对数伪导数定义的加权Sobolev型空间中的稳定性性质。通过引入一个代数缩放参数，我们还引入了基函数的广义版本，这提高了逼近空间的灵活性，并允许更有效地表示奇异因子。误差分析和数值结果表明，所提出的后向对数基函数特别适用于具有终端端点奇异性的弱正则函数，并且能够恢复通常直接应用普通多项式逼近时会丢失的指数或高阶收敛速度。
英文摘要
We introduce a new class of backward logarithmic orthogonal functions and generalized backward logarithmic orthogonal functions, constructed by applying a terminal-endpoint logarithmic mapping to generalized Laguerre polynomials. These functions are designed for backward spectral approximations of problems whose solutions exhibit weak singularities at the terminal endpoint. The proposed basis functions generate non-polynomial weighted approximation spaces with nodes naturally clustered near the singular endpoint, and therefore provide an effective framework for resolving algebraic and logarithmic endpoint singularities. We develop the basic approximation theory for these backward logarithmic orthogonal functions, including recurrence relations, derivative formulas, orthogonality, Sturm--Liouville characterization, mapped Laguerre--Gauss quadrature rules, weighted projection estimates, backward Lagrange interpolation estimates, inverse inequalities, and stability properties in weighted Sobolev-type spaces defined through a terminal logarithmic pseudo-derivative. A generalized version of the basis is also introduced by incorporating an algebraic scaling parameter, which improves the flexibility of the approximation space and allows singular factors to be represented more effectively. The error analysis and numerical results show that the proposed backward logarithmic basis is particularly suitable for weakly regular functions with terminal-endpoint singularities and can recover exponential or high-order convergence rates that are typically lost when usual polynomial approximations are applied directly.
