发表机构
Université de Pau et des Pays de l’Adour (UPPA); University of Calabria; University of Insubria; University of Uppsala（波城与阿杜尔地区大学; 卡拉布里亚大学; 因苏布里亚大学; 乌普萨拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过GLT理论识别Jacobi加权直方插值矩阵的奇异值符号，揭示其分布与条件数特性，并证明不适定性与符号零点集测度相关。
AI 中文摘要
本文首先研究了由Jacobi加权直方插值产生的未缩放矩阵的奇异值分布。对于参数为$(\alpha,\beta)$的Jacobi权重，并使用参数为$(\alpha+1,\beta+1)$的Jacobi基，我们先前证明了当$\alpha,\beta>1/2$且网格为准均匀时，这些矩阵在奇异值分布意义下的增长慢于矩阵尺寸对数的任何正幂次。然而，未缩放矩阵的数值行为暗示存在一个非平凡的奇异值符号，本文明确地识别了该符号。从Jacobi多项式的标准渐近公式出发，我们推导出定义直方插值矩阵条目的加权单元平均值的渐近表示。该表示使我们能够通过GLT理论分析相关的Gram序列。我们识别了其GLT符号，并由此获得了对于任意$\alpha,\beta>-1$的直方插值矩阵的奇异值分布。我们还刻画了所得符号消失的集合$\mathcal Z$并计算其测度，表明该集合仅依赖于网格，而Jacobi参数影响符号在其支撑集上的取值。当该测度为正时，奇异值分布意味着问题的离散渐近不适定性，而与Toeplitz设置的比较表明谱条件数呈指数增长。当$\mathcal Z$的测度为零时，问题适定，且观察到的条件性在矩阵尺寸上呈代数增长。最后，报告并讨论了支持理论发现的数值实验，同时在本工作的最后部分给出了结论及一些开放问题。
英文摘要
In this paper, we first study the singular value distribution of the unscaled matrices arising from Jacobi weighted histopolation. For the Jacobi weight with parameters $(α,β)$, and using the Jacobi basis with parameters $(α+1,β+1)$, we previously proved that, for $α,β>1/2$ and quasi-uniform meshes, the growth of these matrices in the sense of singular value distribution is slower than any positive power of the logarithm of the matrix size. However, the numerical behavior of the unscaled matrices suggested a nontrivial singular value symbol, which we identify explicitly in this paper. Starting from the standard asymptotic formula for Jacobi polynomials, we derive an asymptotic representation of the weighted cell averages defining the entries of the histopolation matrices. This representation allows us to analyze the associated Gram sequence by means of GLT theory. We identify its GLT symbol and, consequently, we obtain the singular value distribution of the histopolation matrices for any $α,β>-1$. We also characterize the set $\mathcal Z$ where the resulting symbol vanishes and compute its measure, showing that this set depends only on the mesh, while the Jacobi parameters affect the values of the symbol on its support. When this measure is positive, the singular value distribution implies the discrete asymptotic ill-posedness of the problem, while a comparison with the Toeplitz setting suggests exponential growth of the spectral condition number. When the measure of $\mathcal Z$ is zero, the problem is well-posed and the observed conditioning is algebraic in the matrix size. Finally, numerical experiments supporting the theoretical findings are reported and discussed, while conclusions are given in the final section of the current work together with a few open problems.
