An arXiv preprint reports a graph-based way to tell when weighted polynomial reconstruction from interval data is uniquely determined. The work studies weighted histopolation on [−1, 1], using weighted integrals over interval cells. Its central fixed-grid result is that unisolvence—the existence of a unique polynomial reconstruction—is equivalent to connectedness of the graph formed by the interval endpoints; whenever the system is unisolvent, that graph is a tree.
For fixed-grid interval families, the moment matrix factors as MS = AS ME, while its Gram matrix is ME^T AS^T AS ME.
A graph test for a unique answer
The graph description also gives an exact infinity-norm condition number—a measure of matrix sensitivity—whenever AS is nonsingular. It equals Wmax(S)Lmax(S): the largest interval width multiplied by the longest path length in the endpoint graph.
For the spectral, or two-norm, condition number, the paper gives a graph-based lower bound involving the same maximum width and maximum path length. That is a bound, not the exact factorization stated for the infinity norm.
Two matrix sequences share the same growth
Two specified sliding-window families provide a more detailed test of matrix behavior. In the nested and supplemented constructions, one interval matrix is related through a diagonal sign matrix to the inverse of the other. Their two-norm condition numbers are the same, and both grow linearly as the matrix size increases.
For N ≥ 3, explicit finite-size singular values are given for both sequences, with the nested values reciprocal to the supplemented values. The common condition number has an explicit trigonometric expression and is asymptotic to (4/π)N as N tends to infinity.
The corresponding limiting singular-value patterns are 2 cos(θ/2) for the supplemented sequence and 1/[2 sin(θ/2)] for the nested sequence.
When the weighted matrix goes diagonal
Another part of the preprint asks when a weighted Gram matrix can be made exactly diagonal. In the first-kind Chebyshev construction, using the stated constant-angular cells, HN^T HN is exactly diagonal, with μ0 = N and explicit formulas for the remaining diagonal entries; HN is nonsingular.
For that construction, κ2(HN) increases strictly with the angular half-length ρ throughout the admissible range. As ρ approaches zero from above, its infimum is the stated √2/a_{N−1} value.
The weight is characterized as well. For N ≥ 3, diagonality for every admissible angular half-length holds if and only if the Jacobi parameters are α = β = −1/2, the first-kind Chebyshev choice.
A broader sufficient criterion uses discrete weighted orthogonality. If the cell-moment columns are nonzero scalar multiples of columns in a discretely WN-orthogonal sampling matrix, the weighted Gram matrix is diagonal and the moment matrix is nonsingular.
The paper also gives a one-column correction rule: a single constant correction to one nonconstant basis element has a unique value that restores a diagonal weighted Gram matrix.
A theorem-first framework
The framework includes a discrete-sine configuration. For the stated nodes and cells, its entries are [MN]ik = (2 sin(kρ)/k)sin(kξi); MN^T MN is diagonal, and MN is nonsingular.
Connected endpoint graphs provide a basis construction too. With positive weights, the induced moment form is an inner product, and Gram–Schmidt yields a unique monic basis whose weighted Gram matrix has a positive diagonal.
For generalized Jacobi weights, the construction defines an orthogonal polynomial family for the modified weight. Its generalized cell moments reduce to finite linear combinations of classical Jacobi cell moments, while positive-degree classical Jacobi moments have an explicit representation in terms of endpoint values.
A related corrected fourth-kind Chebyshev construction uses a different stated range: for N ≥ 2 and 0 < ρ < π/(2N), its Gram matrix has explicit diagonal entries and the matrix is nonsingular.
The boundaries of the result
The study used no empirical sample or dataset. Its conclusions are stated for specified mathematical configurations, including fixed-grid interval families in the graph theorem and the stated weights, cells, nodes, angular ranges and nonsingularity assumptions in the diagonalization results.
The document is an arXiv preprint, version 1, dated 26 Aug 2026.
Paper data and sources
Original title: Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation
Authors: Allal Guessab, Federico Nudo, Stefano Serra-Capizzano
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text