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Conditioning and Singular Value Distribution of Jacobi Weighted Histopolation Matrices

Allal Guessab, Federico Nudo, Stefano Serra-Capizzano

arXiv:2610.04383v1math.NAcs.NA

TL;DR

The singular-value behavior of unscaled Jacobi weighted histopolation matrices was not explained by earlier growth bounds and numerical observations. The paper uses Jacobi-polynomial asymptotics and GLT theory to identify their symbol and zero set. A positive-measure zero set implies discrete asymptotic ill-posedness, while formal conditioning-growth rates remain an open problem.

  • Problem

    Earlier work bounded singular-value-distribution growth only for α, β > 1/2 and quasi-uniform meshes, while unscaled-matrix experiments suggested a nontrivial singular-value symbol.

    Source: §1 · p. 2

  • Method

    Jacobi-polynomial asymptotics provide weighted-cell-average representations for GLT analysis of the associated Gram sequence and identification of its symbol.

    Source: §1 · p. 2, §7 · p. 24

  • Results

    The zero set’s measure depends only on the mesh, and positive measure implies discrete asymptotic ill-posedness for all Jacobi parameters α, β > −1.

    Source: §1 · p. 2, §7 · p. 25

  • Takeaways & Limitations

    When the zero set has measure zero, the problem is well-posed and observed spectral conditioning is algebraic in matrix size.

    Source: §1 · p. 2

    Formal proofs of exponential or algebraic conditioning growth remain open; the authors suggest Toeplitz tools may help.

    Source: §7 · p. 25

Abstract

from arXiv · show

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 $(\alpha,\beta)$, and using the Jacobi basis with parameters $(\alpha+1,\beta+1)$, we previously proved that, for $\alpha,\beta>1/2$ 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 $\alpha,\beta>-1$. 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.

1. Introduction

Weighted histopolation reconstructs from integral data, and earlier analysis left the unscaled matrices’ apparent singular-value behavior unexplained. This paper identifies their singular-value symbol and relates its zero set to ill-posedness and conditioning.

1. Introduction

  • Weighted histopolation addresses reconstruction from data given as weighted averages rather than point evaluations.The reconstruction depends on both the weight and the polynomial basis.Source: §1 · p. 1
  • Earlier work found slower-than-logarithmic-power singular-value growth for α, β > 1/2 on quasi-uniform meshes, while numerical spectra suggested a nontrivial symbol.The present paper is motivated by that behavior in the unscaled matrices.Source: §1 · p. 2
  • Jacobi-polynomial asymptotics yield a representation of weighted cell averages, enabling GLT analysis of the Gram sequence and explicit identification of the singular-value symbol.The resulting analysis gives the singular-value distribution for the unscaled matrices.Source: §1 · p. 2
  • When the symbol’s zero set has positive measure, the singular-value distribution implies discrete asymptotic ill-posedness, and numerical results indicate exponential condition-number growth.The exponential trend is motivated by analogy with the Toeplitz setting.Source: §1 · p. 2
  • When the zero set has measure zero, the problem is well-posed and observed spectral conditioning is algebraic in matrix size.Source: §1 · p. 2

2. Jacobi weighted histopolation matrices and the main result

The section states the main singular-value-distribution result for α, β > −1 under mesh condition (2.1), and characterizes the symbol’s zero set and its mesh dependence.

2. Jacobi weighted histopolation matrices and the main result

  • The mesh determines where the symbol is nonzero, while α and β affect its values on that region.The mesh dependence is expressed through g(y), or equivalently through ϑ(y) and q(y).Source: §2 · p. 3
  • The symbol is positive for |θ| < q(y), and its zero set is Z = {(y, θ): |θ| ≥ q(y)} up to a set of measure zero.The positivity condition is equivalent to the nonemptiness of the sum defining the symbol.Source: §2 · p. 4
  • The cosine mesh is the only admissible mesh with μ0 = 0; every other admissible mesh has μ0 > 0.Source: §2 · p. 5

3. Asymptotic representation

This section derives an asymptotic representation of the histopolation-matrix entries from Jacobi polynomial asymptotics, with uniformity for indices in specified interior ranges.

3. Asymptotic representation

  • The derivation applies the classical Jacobi asymptotic formula, uniform on compact subsets of (0, π), together with changes of variable and mean-value and Taylor estimates.The Jacobi asymptotic formula is used uniformly for y when j ≥ εN.Source: §3 · pp. 6–8
  • An asymptotic representation of the entries of H_N is derived for the weighted cell averages defining the histopolation matrix.The representation is used in the subsequent spectral analysis.Source: §3 · p. 6
  • The entry estimate holds uniformly when ε ≤ y_i ≤ 1 − ε and ε ≤ η_j ≤ 1.These index restrictions keep y_i and η_j in the ranges specified for the representation.Source: §3 · p. 6

4. The Gram sequence

This section establishes a GLT description of the Gram sequence and applies it to the Jacobi weighted histopolation matrices to derive their singular value distribution.

4. The Gram sequence

  • For each fixed diagonal offset, the Gram entries satisfy a uniform asymptotic estimate for indices within the compact interval.This diagonal estimate underpins the fixed-band approximation.Source: §4 · p. 13
  • Fixed-band approximations have Fourier partial-sum symbols converging in L2 to κI, yielding the auxiliary Gram sequence’s GLT symbol.The truncated matrices form an approximating class as the band width grows, and the GLT closure property transfers the limit.Source: §4 · pp. 16–17
  • Proposition 4.2 is applied to determine the GLT symbol of the Gram sequence associated with the Jacobi weighted histopolation matrices {HN}N defined in (2.3).Source: §4 · p. 17
  • The cutoff discrepancy is zero-distributed, so the GLT result for the restricted matrices transfers to the full histopolation sequence.The proof bounds the entries of Q(ε)_N by O(N^-1/2), uniformly in the indices, before concluding zero-distribution.Source: §4 · p. 19
  • The Gram-sequence result is transferred to the singular value distribution of the histopolation matrices using nonnegative singular values.The passage applies the relation between squared singular values and the Gram matrix, then invokes the definition of singular value distribution.Source: §4 · p. 21

5. Further results

The symbol contains information not recoverable from zero-distribution results for scaled matrices. Further results connect its zero set to the mesh and distinguish that dependence from the Jacobi parameters' effect on symbol values.

5. Further results

  • For the uniform mesh, Corollary 5.1 gives the normalized measure of the symbol's zero set.The supplied passage does not state the measure's value.Source: §5 · p. 21
  • The zero set is determined by the mesh, while the Jacobi parameters affect symbol values on its support.For the uniform mesh, the symbol contains information unavailable from scaled-matrix zero-distribution results.Source: §5 · p. 21
  • For the four classical Chebyshev weights, the factor multiplying the symbol's summation has a simple form.The four cases correspond to the first-, second-, third-, and fourth-kind Chebyshev weights.Source: §5 · p. 22

6. Interpretation of the theory and numerical results

Numerical experiments compare unscaled histopolation-matrix singular values with samples of the singular-value symbol across three meshes. The comparisons support the predicted distribution and suggest contrasting conditioning behavior for meshes with positive versus zero-measure symbol zero sets.

6.1. Distribution of the histopolation matrices

  • The experiment compares decreasingly sorted singular values of unscaled histopolation matrices HN with decreasingly sorted samples of the singular-value symbol for the symmetric case, at N = 500, 1000, and 2000.Symbol values are sampled on a uniform tensor grid with 800 points in y and 1600 in θ; matrix entries use Gauss–Legendre quadrature with 105 points per cell.Source: §6.1 · p. 22
  • Figure 1 compares matrix singular values with symbol samples: g1 and g2 decay sharply near their predicted thresholds, while g3 has no analogous interior decay.For g3, the normalized measure of the symbol's nonzero set is 1.Source: Figure 1 · p. 23, §6.1 · pp. 23–24

6.2. Conditioning of the histopolation matrices

  • The singular-value distribution theorem does not determine the smallest singular value, and tiny minima restrict the tested matrix sizes.The final spectrum is intended only as a qualitative indication because roundoff may affect the smallest computed values.Source: §6.1 · p. 22, §6.2 · p. 24
  • κ2(HN) ≃CaN, C > 0, a > 1 for g1 and g2; g3 grows substantially more slowly and suggests algebraic conditioning over the tested range.The reported exponential fits closely follow the computed condition numbers for g1 and g2.Source: §6.2 · p. 24

7. Conclusions

The paper identifies the singular value symbol of the unscaled Jacobi weighted histopolation matrices and characterizes its zero set. The zero set depends only on the mesh, and a positive measure for it implies discrete asymptotic ill-posedness for all α, β > −1.

7. Conclusions

  • The GLT analysis identifies the singular value symbol of the unscaled Jacobi weighted histopolation matrices and characterizes its zero set.The zero set depends only on the mesh, while α and β affect the symbol values on its support.Source: §7 · p. 24
  • When the zero set has positive normalized measure µ0, the problem is discretely asymptotically ill-posed for all α, β > −1.The sharp decay begins near normalized index 1−µ0, the measure of the set where the symbol is nonzero.Source: §7 · p. 25
  • Formal proof of exponential or algebraic conditioning growth, depending on µ0 and the symbol's zero orders, is left as an open problem.The paper suggests tools from the Toeplitz setting could help establish it.Source: §7 · p. 25

Funding statement

The research was supported by GNCS-INdAM 2026 projects, and F. Nudo's work was funded through the EU NextGenerationEU PNRR.

Funding statement

  • The research was supported by GNCS-INdAM 2026 projects, and F. Nudo's work was funded through the EU NextGenerationEU PNRR.The funding statement also identifies the PNRR mission, component, investment, and MUR decree.Source: Funding statement · pp. 25–26

Data Availability

No external datasets were used; the numerical data were generated computationally for the experiments.

Data Availability

  • No external datasets were used; the numerical data were generated computationally for the experiments.Source: Data Availability · p. 26
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