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Exactly Diagonal Gram Matrices in Jacobi Weighted Histopolation

Allal Guessab, Federico Nudo, Stefano Serra-Capizzano

arXiv:2608.25714v1math.NA

TL;DR

Weighted histopolation uses integral data, but its unisolvence and exact Gram-matrix diagonalization depend on interval geometry, weights, and basis choice. This paper characterizes unisolvence through endpoint-graph connectivity and develops orthogonality-based constructions yielding explicit spectral information and diagonal configurations.

  • Problem

    The paper studies how to guarantee unisolvence and exact Gram-matrix diagonality in univariate polynomial weighted histopolation from interval-integral data.

  • Method

    It analyzes endpoint graphs and path-based matrix representations, then uses discrete weighted orthogonality and Jacobi moment reductions to construct diagonalizing polynomial bases.

  • Results

    Unisolvence is equivalent to endpoint-graph connectedness, while the resulting path formulas give explicit singular values and condition numbers, and orthogonality criteria produce exactly diagonal Gram matrices.

  • Takeaways & Limitations

    The framework supplies explicit matrix inverses and spectral descriptions together with diagonal configurations for Chebyshev, constant, connected-graph, and generalized Jacobi settings.

  • Takeaways & Limitations

    Cells that are not Cartesian products are not considered.

Abstract

from arXiv · show

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.

1 Introduction

The paper develops univariate polynomial weighted histopolation on [-1,1] for integral data, linking unisolvence to endpoint-graph structure and exact Gram-matrix diagonality. It combines approximation theory, structured matrices, and asymptotic and numerical linear algebra to derive explicit matrix properties and constructions.

  • Motivation and setting: Histopolation reconstructs functions from average or moment-type integral data over intervals, cells, faces, or more general geometric regions rather than pointwise values.The paper applies this integral-data viewpoint to univariate polynomial weighted histopolation on [-1,1] with Jacobi weights.
  • Endpoint graphs and unisolvence: Unisolvence for interval families on a fixed grid is equivalent to connectedness of the associated endpoint graph.In the unisolvent case, the graph is a tree, and unique paths between consecutive grid points provide an explicit inverse representation.
  • Exact diagonalization: A discrete weighted-orthogonality criterion ensures exact weighted Gram-matrix diagonality, recovering the first-kind Chebyshev construction and producing a constant-weight configuration via discrete sine orthogonality.For connected endpoint graphs, the moment vectors define an inner product and yield a monic basis with diagonal weighted Gram matrix.
  • Scope and organization: The work connects approximation theory, structured matrices, and asymptotic and numerical linear algebra through its unisolvence, diagonalization, and explicit matrix analyses.The paper also derives singular-value and condition-number formulas, limiting singular-value distributions, generalized-Jacobi moment reductions, and shifted-Jacobi basis constructions.

2 A combinatorial unisolvence criterion

For interval families drawn from a fixed grid, weighted histopolation is unisolvent exactly when the endpoint graph is connected. In that case, the graph is a tree, its unique paths yield an explicit inverse representation, and combinatorial path widths control inverse-norm bounds.

  • Graph criterion: Unisolvence of the weighted histopolation problem is equivalent to nonsingularity of the interval matrix AS and, equivalently, to connectedness of the endpoint graph GS.The graph has grid-point vertices and one edge for each interval joining its endpoint indices.
  • Graph criterion: When AS is nonsingular, GS is connected and acyclic, so every pair of grid vertices is joined by a unique path.A connected graph with N+1 vertices and N edges cannot contain a cycle, establishing the tree structure.
  • Inverse representation: The unique paths between grid vertices provide an explicit representation of AS^-1.The inverse construction identifies nonzero entries through the edges of the corresponding path, with signs determined by orientation.
  • Inverse representation: Inverse-norm bounds depend on the interval family through its maximum combinatorial width and the maximum distance between consecutive grid vertices in the endpoint tree.This isolates the family-dependent part of the conditioning estimates from the fixed grid, weight, and polynomial basis.
  • Examples: A disconnected sliding-window family becomes unisolvent after adding one auxiliary elementary interval, which connects its two path components.The original endpoint graph splits into disjoint paths on even and odd vertices; one added interval suffices to restore connectedness.

3 Exactly diagonal weighted histopolation configurations

This section constructs exactly diagonal weighted histopolation configurations for first-kind Chebyshev weights using constant-angular-length cells, and characterizes when this diagonality persists for Jacobi weights. The construction extends to Cartesian product cells in fixed dimensions, while non-Cartesian cells are not considered.

  • Chebyshev configuration: For the first-kind Chebyshev weight, constant-angular-length cells produce a histopolation matrix H_N whose Gram matrix is exactly diagonal.The cells are s_i = [cos(τ_i + ρ), cos(τ_i − ρ)] with 0 < ρ < π/(2N).
  • Chebyshev configuration: The diagonal Gram matrix is positive definite, so H_N is nonsingular and the configuration is unisolvent.The proof establishes positivity of all diagonal entries before concluding nonsingularity.
  • Chebyshev configuration: Within the admissible range, smaller angular cells yield better two-norm conditioning, with H_1 = [1] and κ_2(H_1) = 1.For N ≥ 2, the singular values are given explicitly in the cited remark, and the conditioning improves as ρ decreases.
  • Jacobi parameters: For N ≥ 3, diagonal Gram matrices for every admissible ρ occur exactly when the Jacobi parameters satisfy α = β = −1/2.Theorem 4 states the equivalence between diagonality for every admissible ρ and the first-kind Chebyshev parameters.
  • Cartesian product extension: The exactly diagonal construction extends by tensor products to Cartesian product cells in any fixed dimension, with explicit diagonal Gram entries and singular values.The non-Cartesian-cell case is explicitly outside the scope of the section.

4 A general diagonalization criterion

The section develops a discrete weighted-orthogonality criterion that makes histopolation Gram matrices diagonal and guarantees nonsingularity. It applies the criterion to Chebyshev and discrete sine constructions, and connects endpoint-graph connectedness with positive-definite diagonal Gram matrices.

  • 4 A general diagonalization criterion: The criterion recovers the first-kind Chebyshev construction without correction and produces an exactly diagonal configuration from discrete sine orthogonality.For the sine-based construction, the unnormalized histopolation matrix is nonsingular and its columns are scalar multiples of sampled sine vectors.
  • 4 A general diagonalization criterion: If weighted cell moments are nonzero scalar multiples of W_N-orthogonal sampled vectors, then M_N^T W_N M_N is diagonal and M_N is nonsingular.The factorization M_N = Ψ_NΓ_N transfers discrete orthogonality to the histopolation matrix.
  • 4 A general diagonalization criterion: Correcting one polynomial column under the stated scalar conditions yields a diagonal weighted Gram matrix and a nonsingular modified histopolation matrix.The correction subtracts the W_N-orthogonal projection onto the span of the first column, and its coefficient is uniquely determined.
  • 4 A general diagonalization criterion: The diagonal Gram identity determines the singular values of W_N^1/2 M̂_N, with the usual singular values recovered when W_N = I_N.The singular values are given by the square roots of the diagonal Gram entries, up to ordering.
  • 4 A general diagonalization criterion: Connectedness of the endpoint graph G_S is equivalent to the induced moment form being an inner product, and it yields a polynomial basis with positive-definite diagonal weighted Gram matrix.If G_S is disconnected, the associated matrix is singular, so the moment form is not an inner product.

5 Generalized Jacobi weights and moment reduction

The section relates generalized Jacobi histopolation to the classical Jacobi setting through polynomial correction and finite moment reductions. It also develops the ingredients for an exactly diagonal configuration for the Chebyshev weight of the fourth kind.

  • Chebyshev construction: The generalized Jacobi reduction is used to construct an exactly diagonal configuration for the Chebyshev weight of the fourth kind.The section explicitly states this construction as one of its aims.
  • Generalized Jacobi weights: The constructed polynomial family is monic and orthogonal with respect to the generalized Jacobi weight, with coefficients uniquely determined by a nonsingular system.The proof establishes uniqueness through the triviality of the corresponding homogeneous system.
  • Moment reduction: Generalized Jacobi cell moments are expressed as finite linear combinations of classical Jacobi cell moments.This provides the stated reduction from generalized Jacobi histopolation to classical Jacobi histopolation.
  • Moment reduction: Positive-degree Jacobi cell moments admit explicit representations using endpoint values.The representation follows by integrating a weighted differentiation identity over each cell.
  • Polynomial correction: A correction construction uniquely associates modified polynomials with Jacobi polynomials of degrees below N, and the resulting quotient family forms a basis of ΠN−1.The correction subtracts a given Jacobi polynomial so the remainder is a multiple of the prescribed polynomial factor.

The quotient

The quotient construction produces polynomial basis elements that are linearly independent, while a corrected Chebyshev fourth-kind configuration yields a nonsingular moment matrix with a diagonal Gram matrix.

  • The quotient: The quotient in (64) is a polynomial, and the resulting polynomials eQ0, ..., eQN−1 form a basis of ΠN−1.The quotient follows because χp,γ+,γ− divides the relevant numerator; linear independence then establishes the basis property.
  • The quotient: Rescaling the nonconstant quotient elements by 1 −ζx preserves a basis of ΠN−1 and permits moment entries to be obtained explicitly from cell endpoints.The endpoint condition k(ζ) ≠ 0 is used for k = 1, ..., N −1.
  • Chebyshev fourth-kind configuration: For the first-kind Chebyshev weight, the shifted weight is the Chebyshev weight of the fourth kind, and the basis functions produce sampled cosine vectors.The key identity is k(cos θ)ω1/2,−1/2(cos θ) sin θ = cos(Nθ)−cos(kθ).
  • Chebyshev fourth-kind configuration: The sampled cosine vectors are mutually orthogonal, enabling a correction of one nonconstant element that yields a diagonal Gram matrix and a nonsingular moment matrix.The corrected matrix has positive definite ˆM_N^⊤ˆM_N, hence is nonsingular.

6 Conclusions and Future Work

The work characterizes unisolvence through endpoint-graph connectedness and develops diagonal Gram-matrix constructions for weighted histopolation. It also identifies open problems concerning orthogonal bases, tree topology, and generalized Jacobi weights.

  • Conclusions: Unisolvence is equivalent to connectedness of the endpoint graph; in that case, the graph has N + 1 vertices and N edges and is a tree.This characterization applies to interval families whose endpoints belong to a fixed grid.
  • Conclusions: A discrete weighted-orthogonality criterion yields diagonalization results, recovers the first kind Chebyshev construction, and includes constant-weight configurations based on discrete sine orthogonality.The results also include a correction for a single nonconstant column.
  • Conclusions: For connected endpoint graphs, a polynomial basis was constructed whose corresponding weighted Gram matrix is diagonal.This extends the diagonalization framework to interval families with connected endpoint graphs.
  • Future Work: Future work includes explicit formulas or recurrence relations for the graph-associated orthogonal basis and a more precise analysis of how tree topology affects diagonal entries and the moment-matrix spectral condition number.These questions concern the orthogonal basis associated with the endpoint graph and the corresponding tree.
  • Future Work: Future work will also seek explicit orthogonal bases for connected endpoint graphs using moment reduction formulas for generalized Jacobi weights.The proposed direction applies the reduction formulas to interval families with connected endpoint graphs.

Declarations · Corresponding author · Funding statement

The funding statement identifies GNCS-INdAM 2026 project support and European Union–NextGenerationEU funding for F. Nudo under Italy’s PNRR.

  • Funding statement: GNCS-INdAM 2026 supported the research through two projects on polynomial methods, kernels, approximation software, structured methods, and advanced signal processing.The projects are titled “Metodi polinomiali e kernel per l’approssimazione da dati discreti e integrali con software OS” and “Metodi strutturati per il signal processing avanzato”.
  • Funding statement: The GNCS-INdAM support is associated with CUP E53C25002010001.The identifier appears at the end of the GNCS-INdAM funding acknowledgment.
  • Funding statement: F. Nudo’s work was funded by the European Union through NextGenerationEU.The funding is stated separately from the GNCS-INdAM project support.
  • Funding statement: The European Union funding was provided under Italy’s National Recovery and Resilience Plan (PNRR).The acknowledgment specifies the PNRR framework for the NextGenerationEU support.
  • Funding statement: The funding falls under PNRR Mission 4, Component 2, Investment 1.2.Investment 1.2 is identified as “Finanziamento di progetti presentati da giovani ricercatori”.
  • Funding statement: The acknowledgment states that the funding was provided pursuant to an MUR Decree.The supplied passage ends after “MUR Decree”, without giving the decree number or further details.
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