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Gauss–Jacobi nodes and weights are well-defined

Statement

Assume Countable Choice. For $n \geq 1$ and $α, β > - 1$ , the measure and order- $2 n - 1$ moment datum in The Gauss–Jacobi quadrature rule are valid Gaussian quadrature data. In particular the monic orthogonal polynomial exists, its $n$ roots are distinct points of $(- 1, 1)$ , and the Lagrange weights and finite rule are well-defined.

Facts & Assumptions

Given: Countable Choice, an integer $n \geq 1$ , and real $α, β > - 1$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed for construction of the Jacobi measure.

[F1]

The Jacobi weight defines a finite positive measure with full support: $μ_{α, β}$ is a finite positive Borel measure whose relative support is all of $[- 1, 1]$ and whose moments of every finite degree are finite.

[F2]

Positive-measure Gaussian quadrature moment data: order- $2 n - 1$ moment data require a finite positive measure, at least $n + 1$ support points, and finite absolute moments through degree $2 n - 1$ .

[F3]

The Gauss–Jacobi quadrature rule and The m-node Gaussian quadrature rule for a positive measure: the Jacobi rule is the general Gaussian rule for this moment datum, with roots of its monic orthogonal polynomial as nodes and Lagrange-integral weights.

[F4]

The Gaussian nodes, cardinal polynomials and weights are well-defined: for order- $2 n - 1$ Gaussian moment data, the nodes can be ordered as distinct interior roots, each Lagrange polynomial is defined, every weight is finite, and the rule is a finite real sum.

Proof

1.1A1F1F2givenalgebraconstruct

By [F1], the measure is finite and positive, all required moments are finite, and each of the $n + 1$ explicit points $t_{j} = - 1 + 2 j / n$ for $j = 0, \dots, n$ lies in its support; these points are distinct because $n \geq 1$ . Thus the data satisfy [F2].

2.1F3F4step 1.1∎

Applying [F4] to the moment datum of step 1.1 gives distinct roots in $(- 1, 1)$ , nonzero node differences, finite Lagrange-weight integrals, and a finite rule. These are exactly the asserted well-definedness properties in [F3].

Depends on

Used by

Gauss–Jacobi exactness and the derivative remainder Theorem

Cited to discharge well-definedness by The Gauss–Jacobi quadrature rule.

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Sources

Christopher J. Bishop, The Riemann Mapping Theorem, Chapter 3 §3, Gauss-Jacobi quadrature lecture notes (standard reference, not scraped)