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Two Gaussian weights are the squared norms of the two cardinal polynomials

Example

Let $(a, b, 2, 3, μ)$ be Gaussian moment data, with distinct Gaussian nodes $x_{1} < x_{2}$ and weights $w_{1}, w_{2}$ . Its cardinal polynomials are $ℓ_{1} (X) = \frac{X - x_{2}}{x_{1} - x_{2}}, ℓ_{2} (X) = \frac{X - x_{1}}{x_{2} - x_{1}} .$ The degree-three exactness theorem implies $w_{1} = \int ℓ_{1}^{2} d μ = \frac{μ_{2} - 2 x_{2} μ_{1} + x_{2}^{2} μ_{0}}{(x_{1} - x_{2})^{2}} > 0, w_{2} = \int ℓ_{2}^{2} d μ = \frac{μ_{2} - 2 x_{1} μ_{1} + x_{1}^{2} μ_{0}}{(x_{2} - x_{1})^{2}} > 0,$ where $μ_{j} = \int x^{j} d μ$ for $j = 0, 1, 2$ .

Facts & Assumptions

Given: Moment data of order $3$ , its two-node Gaussian rule, and its distinct nodes $x_{1}, x_{2}$ .

[F1]

The m-node Gaussian quadrature rule for a positive measure: $w_{i} = \int ℓ_{i} d μ$ and $Q_{2} (f) = w_{1} f (x_{1}) + w_{2} f (x_{2})$ .

[F2]

Gaussian quadrature is exact through degree 2m−1 and has positive weights: the rule is exact through degree $3$ and $w_{i} > 0$ .

[F3]

Cardinal-polynomial interpolation and Cardinal interpolation functions are polynomials of the stated degrees: the displayed $ℓ_{i}$ are degree one and satisfy $ℓ_{i} (x_{k}) = 1$ for $k = i$ and $0$ otherwise.

[F4]

Positive-measure Gaussian quadrature moment data and The Lebesgue integral is linear on $L^{1} (μ)$ : for $j = 0, 1, 2$ , $\int x^{j} d μ = μ_{j}$ , and integration distributes over the finite polynomial expansion.

Proof

1.1F1F2F3

Since $\deg (ℓ_{i}^{2}) = 2$ , exactness [F2] gives $Q_{2} (ℓ_{i}^{2}) = \int ℓ_{i}^{2} d μ$ . Expanding the finite rule sum [F1] and using the cardinal values [F3] gives $Q_{2} (ℓ_{i}^{2}) = w_{i}$ , hence $\int ℓ_{i}^{2} d μ = w_{i}$ .

2.1F2F4step 1.1algebra∎

The formula for $ℓ_{1}$ gives $ℓ_{1} (x)^{2} = (x^{2} - 2 x_{2} x + x_{2}^{2}) / (x_{1} - x_{2})^{2}$ . Integrating term by term using [F4] proves the displayed expression for $w_{1}$ . Replacing the omitted node $x_{2}$ by $x_{1}$ gives the displayed expression for $w_{2}$ . Their strict positivity is [F2].

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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)