Spectral distribution of Jacobi weighted histopolation matrices via GLT theory
For researchers in numerical linear algebra and approximation theory, this work provides a theoretical analysis of matrix structures arising from weighted histopolation, but the results are incremental as they extend known GLT theory to a specific problem.
The paper studies a weighted histopolation problem on [-1,1] with Jacobi weights, deriving exact factorizations of histopolation matrices and showing that the resulting matrix sequences belong to the GLT class, which yields spectral distributions and insights into numerical stability.
In this paper we study a weighted histopolation problem on $[-1,1]$ associated with Jacobi weights. In the first part of the present work we prove results in approximation theory, while in the second we analyze the resulting matrices from an asymptotic linear algebra perspective. More in detail, in the first part, given weighted cell averages, we construct a reconstruction operator based on weighted primitives of Jacobi polynomials and investigate the resulting discretization matrices. At any fixed discretization level, we derive an exact factorization of the histopolation matrix through a backward-difference operator and a sampling operator of Jacobi weighted primitives. Combining a sharp integration by parts identity with the three-term recurrence of Jacobi polynomials, we further show that the primitive sampling operator admits an explicit decomposition involving a tridiagonal coupling matrix in the Jacobi spectral index. This yields a tridiagonal factor representation of the histopolation matrix. In the second part, under standard mesh-regularity assumptions, we show that all the various induced matrix sequences belong to the Generalized Locally Toeplitz (GLT) class, by describing in detail the related GLT symbols. As a consequence, we provide the corresponding spectral distributions and discuss their implications for numerical stability when solving the associated linear systems.