AI 中文总结
研究基于反高斯雅可比多项式零点的新拉格朗日插值过程，建立其节点性质及相关估计，构建加权勒贝格常数对数增长的插值过程，相比已知方案有优势，还给出收敛估计且实验支持理论。
AI 中文摘要
本文介绍并研究了一种基于反高斯雅可比多项式零点的新拉格朗日插值过程。建立了反高斯节点的基本性质，包括渐近分布，以及相关多项式及其导数的估计。这些结果为构建加权勒贝格常数呈对数增长的插值过程提供了基础，确保了最优逼近性质。与基于雅可比节点的先前已知插值方案相比，该过程在端点权重参数的偏移范围内实现了最优勒贝格常数，允许使用较小的端点权重指数。为合适的加权索伯列夫空间中的函数建立了收敛估计，数值实验支持了理论结果。
英文摘要
This paper introduces and investigates a new Lagrange interpolation process based on the zeros of anti-Gauss Jacobi polynomials. Fundamental properties of anti-Gauss nodes, including their asymptotic distribution, are established, together with estimates for the associated polynomials and their derivatives. These results provide the basis for the construction of an interpolation process whose weighted Lebesgue constants exhibit logarithmic growth, ensuring optimal approximation properties. Compared with previously known interpolation schemes based on Jacobi nodes, the proposed process achieves optimal Lebesgue constants for a shifted range of endpoint weight parameters, allowing the use of smaller endpoint weight exponents. Convergence estimates are established for functions in suitable weighted Sobolev spaces, and numerical experiments support the theoretical findings.
