LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The Gaussian nodes, cardinal polynomials and weights are well-defined

Statement

For moment data with order $2 m - 1$ , the nodes and rule defined in The m-node Gaussian quadrature rule for a positive measure exist as stated: the roots can be ordered in $(a, b)$ , every cardinal polynomial is defined, every weight integral is finite, and $Q_{m} (f)$ is a finite real number whenever $f$ is real-valued at the nodes.

Facts & Assumptions

Given: Moment data $(a, b, m, 2 m - 1, μ)$ and the rule of The m-node Gaussian quadrature rule for a positive measure.

[F1]

Positive-measure Gaussian quadrature moment data: moments through degree $2 m - 1$ are finite.

[F2]

Monic orthogonal polynomials exist uniquely through the quadrature order and The Gauss orthogonal polynomial has simple roots inside the interval: $p_{m}$ is monic and has exactly $m$ distinct roots in $(a, b)$ .

[F3]

Cardinal-polynomial interpolation: for distinct nodes, $ℓ_{i} (X) = \prod_{j \neq i} (X - x_{j}) / (x_{i} - x_{j})$ ; every denominator is a nonzero node difference.

[F4]

Cardinal interpolation functions are polynomials of the stated degrees: each $ℓ_{i}$ is a polynomial of degree at most $m - 1$ and has the cardinal values at the nodes.

[F5]

The Lebesgue integral is linear on $L^{1} (μ)$ : a finite linear combination of integrable monomials is integrable and its integral is the corresponding finite linear combination of moments.

Proof

1.1F2F3F4construct

By [F2], the roots of $p_{m}$ form a finite set of exactly $m$ distinct points in $(a, b)$ , so sorting them defines $x_{1} < \dots < x_{m}$ without ambiguity. Their pairwise differences are nonzero, so [F3] defines every $ℓ_{i}$ . By [F4], each cardinal polynomial has degree at most $m - 1$ .

1.2F1F4F5

Each $ℓ_{i}$ is a finite linear combination of monomials of degrees at most $m - 1$ . These moments are finite by [F1], so [F5] makes every $w_{i} = \int ℓ_{i} d μ$ finite.

2.1F1step 1.2algebra∎

For any real values $f (x_{1}), \dots, f (x_{m})$ , the expression $\sum_{i} w_{i} f (x_{i})$ has finitely many finite real summands, hence is a defined real number. This establishes the stated well-definedness of $Q_{m} (f)$ .

Depends on

Used by

Cited to discharge well-definedness by The m-node Gaussian quadrature rule for a positive measure.

Dependency tree · two levels

44 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)