雅可比加权直方图插值中的严格对角格拉姆矩阵
Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation
浏览论文内容
中文总结 AI 辅助
该研究探讨雅可比加权直方图插值，通过端点图连通性表征单可解性，建立对角化准则，推导单元矩约化公式，得到严格对角格拉姆矩阵及奇异值、条件数等结果。
中文摘要 AI 辅助
在本研究中，我们探讨区间[-1,1]上的单变量多项式加权直方图插值问题，其中数据为一系列区间上的加权积分。选定多项式基后，加权矩条件会生成一个直方图插值矩阵，其结构取决于权重和单元几何形状。我们研究该矩阵的非奇异性（它保证了单可解性）及其格拉姆矩阵的严格对角性，后者可明确确定其奇异值和谱条件数。对于端点属于固定网格的区间族，我们通过关联端点图的连通性来表征单可解性；在单可解情形下，该图为树，连接连续网格点的唯一路径可给出逆矩阵的显式表达式。此恒等式为两个矩阵的奇异值提供了显式公式，表明它们的二范数条件数一致且随矩阵规模线性增长，还得到了两个矩阵序列的极限奇异值分布。我们还建立了基于离散加权正交性的一般对角化准则，该准则可恢复第一类切比雪夫构造，并基于离散正弦正交性为常数权重生成对角配置。对于端点图连通的区间族，对应的矩向量定义了多项式空间上的内积，并生成具有对角加权格拉姆矩阵的首一基。最后，我们推导了与广义雅可比权重相关的单元矩的约化公式，并引入了移位雅可比权重的替代基；将该基应用于第四类切比雪夫权重，并修正一个非常数元素，可得到严格对角的格拉姆矩阵。
英文摘要
In the current work, we study univariate polynomial weighted histopolation on $[-1,1]$, where the data are weighted integrals over a family of intervals. After choosing a polynomial basis, the weighted moment conditions lead to a histopolation matrix whose structure depends on the weight and on the geometry of the cells. We investigate its nonsingularity, which guarantees unisolvence, together with exact diagonality of its Gram matrix, which allows its singular values and spectral condition number to be determined explicitly. For families of intervals whose endpoints belong to a fixed grid, we characterize unisolvence in terms of the connectedness of the associated endpoint graph. In the unisolvent case, this graph is a tree, and the unique paths joining consecutive grid points provide an explicit expression for the inverse matrix. This identity gives explicit formulas for the singular values of both matrices, and shows that their condition numbers in the two-norm coincide and grow linearly with the matrix size. Moreover, it yields the limiting singular value distributions of the two matrix sequences. We also establish a general diagonalization criterion based on discrete weighted orthogonality. The criterion recovers the first kind Chebyshev construction and leads to a diagonal configuration for the constant weight based on discrete sine orthogonality. For interval families with a connected endpoint graph, the corresponding moment vectors define an inner product on the polynomial space and lead to a monic basis with a diagonal weighted Gram matrix. Finally, we derive reduction formulas for cell moments associated with generalized Jacobi weights and introduce an alternative basis for shifted Jacobi weights. Applied to the Chebyshev weight of the fourth kind, this basis, together with a correction of one nonconstant element, yields an exactly diagonal Gram matrix.
