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A residual and spectral gap bound a Gaussian weight error

Statement

Let $(a, b, m, 2 m - 1, μ)$ be moment data. Put $μ_{0} : = μ ([a, b]) > 0$ . Let $J$ be the $m \times m$ real symmetric Jacobi matrix of The Jacobi matrix spectrum is the set of Gauss nodes, with distinct eigenvalues $λ_{1} < \dots < λ_{m}$ and Gaussian weights $w_{i}$ of Gaussian weights equal the total mass times a Jacobi eigenvector's first-coordinate square. Fix $i$ , a nonzero vector $v \in R^{m}$ , a real number $θ$ , and $ρ \geq 0$ . Suppose $∥ v ∥_{2} \geq s > 0$ and $∥ J v - θ v ∥_{2} \leq ρ, g : = \min_{k \neq i} ∣ λ_{k} - θ ∣ > 0.$ For $m \geq 2$ , $∣ w_{i} - μ_{0} \frac{v_{1}^{2}}{∥ v ∥_{2}^{2}} ∣ \leq \frac{2 \sqrt{2} μ_{0} ρ}{s g} .$

Facts & Assumptions

Given: Moment data, a real symmetric $J$ with distinct ordered eigenvalues, its $i$ th Gaussian weight, and a nonzero candidate vector $v$ and shift $θ$ satisfying the stated residual and gap bounds.

[F1]

Real spectral theorem: a self-adjoint endomorphism of a finite-dimensional real inner product space has an orthonormal eigenbasis: $J$ has an orthonormal eigenbasis $u_{1}, \dots, u_{m}$ with $J u_{k} = λ_{k} u_{k}$ .

[F2]

Positive-measure Gaussian quadrature moment data: $μ_{0} = μ ([a, b]) > 0$ .

[F3]

Gaussian weights equal the total mass times a Jacobi eigenvector's first-coordinate square: $w_{i} = μ_{0} (u_{i})_{1}^{2}$ for a unit eigenvector $u_{i}$ at $λ_{i}$ .

[F4]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : on $R^{m}$ , $∥ x ∥_{2}^{2} = \sum_{k = 1}^{m} x_{k}^{2}$ and each summand is nonnegative.

Proof

1.1givenF1F4algebra

Normalize $\hat{v} : = v / ∥ v ∥_{2}$ . Then $∥ J \hat{v} - θ \hat{v} ∥_{2} = ∥ J v - θ v ∥_{2} / ∥ v ∥_{2} \leq ρ / s$ . Expand $\hat{v} = \sum_{k = 1}^{m} c_{k} u_{k}$ in the orthonormal basis [F1]. Then $∥ J \hat{v} - θ \hat{v} ∥_{2}^{2} = \sum_{k = 1}^{m} c_{k}^{2} (λ_{k} - θ)^{2} \geq g^{2} \sum_{k \neq i} c_{k}^{2} .$ Thus $\sum_{k \neq i} c_{k}^{2} \leq ρ^{2} / (s^{2} g^{2})$ .

2.1givenstep 1.1F1F4algebra

Put $ε = 1$ if $c_{i} \geq 0$ and $ε = - 1$ otherwise. Since $\hat{v}$ and $u_{i}$ are unit vectors, $∥ \hat{v} - ε u_{i} ∥_{2}^{2} = 2 (1 - ∣ c_{i} ∣) \leq 2 (1 - c_{i}^{2}) = 2 \sum_{k \neq i} c_{k}^{2} \leq 2 ρ^{2} / (s^{2} g^{2})$ .

3.1F4step 2.1algebra

By [F4], ${\hat{v}}_{1}^{2} \leq \sum_{k = 1}^{m} {\hat{v}}_{k}^{2} = 1$ and $(u_{i})_{1}^{2} \leq \sum_{k = 1}^{m} (u_{i})_{k}^{2} = 1$ , since all coordinate squares are nonnegative. Hence $∣ {\hat{v}}_{1} ∣, ∣ (u_{i})_{1} ∣ \leq 1$ . Therefore $∣ {\hat{v}}_{1}^{2} - (u_{i})_{1}^{2} ∣ = ∣ {\hat{v}}_{1} - ε (u_{i})_{1} ∣ ∣ {\hat{v}}_{1} + ε (u_{i})_{1} ∣ \leq 2 ∥ \hat{v} - ε u_{i} ∥_{2} \leq 2 \sqrt{2} ρ / (s g) .$

4.1F2F3step 3.1algebra∎

Since ${\hat{v}}_{1}^{2} = v_{1}^{2} / ∥ v ∥_{2}^{2}$ , multiply step 3.1 by $μ_{0} > 0$ from [F2] and use [F3]. The result is the claimed Gaussian weight bound.

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Sources

John A. Gubner, Gaussian Quadrature and the Eigenvalue Problem, §§2–5, 2020 (standard reference, not scraped)
MATH 373 Lecture Notes, Lecture 14: Gaussian Quadrature, §§10.2–10.5 (standard reference, not scraped)
Walter Gautschi, The Interplay Between Classical Analysis and (Numerical) Linear Algebra, §2.1 (standard reference, not scraped)