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The polynomial moment pairing is a real inner product

Statement

Let $(a, b, m, r, μ)$ be moment data as in Positive-measure Gaussian quadrature moment data. For $0 \leq k \leq m$ with $2 k \leq r$ , the pairing The polynomial moment pairing is a well-defined real inner product on $R^{k + 1}$ .

Facts & Assumptions

Given: Moment data $(a, b, m, r, μ)$ , an integer $k$ with $0 \leq k \leq m$ and $2 k \leq r$ , and coefficient vectors $c, d \in R^{k + 1}$ .

[F1]

Positive-measure Gaussian quadrature moment data: $μ$ is a finite positive Borel measure; its relative support contains at least $m + 1$ points; and $∣ x ∣^{j}$ is integrable for every $j \leq r$ .

[F2]

The polynomial moment pairing: $P_{c} (x) = \sum_{i = 0}^{k} c_{i} x^{i}$ and $⟨ c, d ⟩_{μ} = \int P_{c} P_{d} d μ$ .

[F3]

A nonzero real polynomial of degree $n$ has no more than $n$ distinct real roots: a nonzero real polynomial of degree at most $k$ has at most $k$ distinct real roots.

[F4]

Monotonicity and nonnegative homogeneity of the nonnegative integral, The nonnegative integral agrees with the simple integral on simple functions, and The integral of a nonnegative simple function: the nonnegative integral is monotone and positively homogeneous, and $\int 1_{U} d μ = μ (U)$ for a Borel set $U$ .

[F5]

The Lebesgue integral is linear on $L^{1} (μ)$ : finite sums and real scalar multiples of integrable functions are integrable and their integrals obey the corresponding linearity rules.

[F6]

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: every polynomial function is continuous; the relative Borel sigma-algebra is generated by the relative open sets (The Borel sigma-algebra of a topological space).

[F7]

The reals form a field: real addition and multiplication are distributive and multiplication is commutative.

[F8]

Real and complex inner product spaces, with the inner product linear in the first argument: a real inner product is bilinear, symmetric, nonnegative on the diagonal, and definite.

Proof

1.1F1F2F5algebra

Each monomial $x^{j}$ with $j \leq 2 k$ is integrable by [F1]. Expanding the finite coefficient sums in [F2] expresses $P_{c} P_{d}$ as a finite real linear combination of such monomials, so [F5] makes the product integrable. Distributivity and [F5] then give bilinearity in each coefficient vector.

1.2F2F7algebra

Real multiplication is commutative by [F7], so $P_{c} P_{d} = P_{d} P_{c}$ pointwise and the pairing is symmetric.

1.3F1F2F3choose

If $c \neq 0$ , its coefficient list defines a nonzero formal polynomial $P_{c}$ of degree at most $k$ . By [F3], its zero set has at most $k$ points. Since the support in [F1] has at least $m + 1 \geq k + 1$ distinct points, choose a support point $s$ with $P_{c} (s) \neq 0$ .

2.1F1F2F4F6step 1.3choosealgebra

Put $α = P_{c} (s)^{2} > 0$ . Continuity in [F6] gives a relative open neighborhood $U$ of $s$ on which $P_{c} (x)^{2} > α / 2$ . This set is Borel and has $μ (U) > 0$ by the support definition. Monotonicity, homogeneity and the indicator identity in [F4] imply $⟨ c, c ⟩_{μ} = \int P_{c}^{2} d μ \geq \frac{α}{2} μ (U) > 0.$

3.1step 1.1step 1.2step 2.1F8∎

The pairing is well-defined by step 1.1, real-bilinear and symmetric by steps 1.1–1.2, nonnegative on the diagonal, and zero on the diagonal only for $c = 0$ by step 2.1. These are the real inner product axioms in [F8].

Depends on

Used by

Cited to discharge well-definedness by The polynomial moment pairing.

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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)