TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Levin endpoint identity and conditioned residual bound

Statement

Let $a < b$ , $s \geq 1$ , $ω \in R ∖ {0}$ , complex-valued $f \in C^{s} [a, b]$ , and real-valued $g \in C^{s + 1} [a, b]$ with $∣ g^{'} ∣ \geq c > 0$ . Use a nonsingular collocation system from Levin collocation for a nonstationary oscillatory integral, with complex-valued basis functions $ψ_{k} : [a, b] \to C$ in $C^{s + 1} [a, b]$ . Write $A_{j k} = L_{ω} ψ_{k} (x_{j})$ , $y_{j} = f (x_{j})$ , $c = A^{- 1} y$ , $p = \sum_{k} c_{k} ψ_{k}$ , $Q = p (b) e^{i ω g (b)} - p (a) e^{i ω g (a)}$ , and $r = f - L_{ω} p$ . Then $I_{ω} (f, g; [a, b]) - Q = I_{ω} (r, g; [a, b]) = \sum_{j = 0}^{s - 1} \frac{(- 1)^{j}}{(i ω)^{j + 1}} {[\frac{T_{g}^{j} r (x)}{g^{'} (x)} e^{i ω g (x)}]}_{a}^{b} + \frac{(- 1)^{s}}{(i ω)^{s}} I_{ω} (T_{g}^{s} r, g; [a, b]) .$ Put $E_{j} = ∣ T_{g}^{j} f (a) / g^{'} (a) ∣ + ∣ T_{g}^{j} f (b) / g^{'} (b) ∣$ , $V_{j} = ∥ v_{j} (a) ∥_{2} + ∥ v_{j} (b) ∥_{2}$ for $v_{j, k} (x) = T_{g}^{j} (L_{ω} ψ_{k}) (x) / g^{'} (x)$ , and $K_{s} = (\sum_{k} ∥ T_{g}^{s} (L_{ω} ψ_{k}) ∥_{\infty}^{2})^{1 / 2}$ . Then $∣ I_{ω} (f, g) - Q ∣ \leq \sum_{j = 0}^{s - 1} \frac{E_{j} + ∥ A^{- 1} ∥_{2} ∥ y ∥_{2} V_{j}}{∣ ω ∣^{j + 1}} + \frac{b - a}{∣ ω ∣^{s}} (∥ T_{g}^{s} f ∥_{\infty} + ∥ A^{- 1} ∥_{2} ∥ y ∥_{2} K_{s}) .$ If an approximate coefficient vector $\hat{c}$ has exact residual $ρ = A \hat{c} - y$ and endpoint vector $d_{k} = ψ_{k} (b) e^{i ω g (b)} - ψ_{k} (a) e^{i ω g (a)}$ , its output satisfies $∣ Q (\hat{c}) - Q ∣ \leq ∥ d ∥_{2} ∥ A^{- 1} ∥_{2} ∥ ρ ∥_{2} = ∥ d ∥_{2} \frac{κ_{2} (A)}{∥ A ∥_{2}} ∥ ρ ∥_{2} .$ If $B_{s}$ denotes the right-hand side of the preceding collocation bound, then $∣ I_{ω} (f, g) - Q (\hat{c}) ∣ \leq B_{s} + ∥ d ∥_{2} ∥ A^{- 1} ∥_{2} ∥ ρ ∥_{2} .$ All vector norms are Euclidean, $∥ A ∥_{2} = \sup_{∥ v ∥_{2} = 1} ∥ A v ∥_{2}$ is the induced matrix norm, and $κ_{2} (A) = ∥ A ∥_{2} ∥ A^{- 1} ∥_{2}$ . Thus a small nodal residual alone does not provide an integral-error bound; the smooth residual terms and inverse solve amplification are also required.

Facts & Assumptions

Given: The stated functions, nonzero frequency, nonstationary phase, nonsingular collocation matrix, and exact arithmetic residual $ρ$ when an approximate coefficient vector is considered.

[F1]

Levin collocation for a nonstationary oscillatory integral: $L_{ω} p = p^{'} + i ω g^{'} p$ , $A_{j k} = L_{ω} ψ_{k} (x_{j})$ , and the endpoint output is $p (b) e^{i ω g (b)} - p (a) e^{i ω g (a)}$ .

[F2]

Derivative of a complex oscillatory phase: $(e^{i ω g})^{'} = i ω g^{'} e^{i ω g}$ .

[F3]

If $u, v$ are differentiable on $[a, b]$ with $u^{'}, v^{'}$ integrable, then $\int_{a}^{b} u v^{'} = u (b) v (b) - u (a) v (a) - \int_{a}^{b} u^{'} v$ : for real $C^{1}$ functions, $\int u v^{'} = [u v] - \int u^{'} v$ ; the complex identity follows componentwise.

[F4]

Repeated integration by parts for a nonstationary oscillatory integral: the $s$ -step expansion and its remainder bound apply to every $C^{s}$ amplitude when $∣ g^{'} ∣ \geq c > 0$ .

[F5]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: for finite real or complex vectors, $∣ ⟨ z, w ⟩ ∣ \leq ∥ z ∥_{2} ∥ w ∥_{2}$ .

[F6]

Real Riemann integrals on an interval are linear by Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ with $m = 1$ ; the componentwise complex convention Finite-interval oscillatory integral and phase convention extends this to complex integrands.

[F7]

The complex modulus is subadditive, $∣ z + w ∣ \leq ∣ z ∣ + ∣ w ∣$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[F8]

On $C^{n}$ the Euclidean inner product is $⟨ x, y ⟩ = \sum_{k} x_{k} \overline{y_{k}}$ , linear in the first argument, and $∥ \overline{d} ∥_{2} = ∥ d ∥_{2}$ by Real and complex inner-product spaces and their induced length and Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive.

Proof

1.1givenF1F2F3F6algebra

For any $p \in C^{1} [a, b]$ , [F2] gives $(p e^{i ω g})^{'} = (L_{ω} p) e^{i ω g}$ . Apply [F3] to the real and imaginary components of $p$ and $e^{i ω g}$ ; the product rule and cancellation of the two $i ω g^{'} p$ terms yield $I_{ω} (L_{ω} p, g) = [p e^{i ω g}]_{a}^{b}$ . With $p$ from [F1] and $r = f - L_{ω} p$ , componentwise linearity [F6] gives $I_{ω} (f, g) - Q = I_{ω} (r, g)$ .

1.2F1givenalgebra

The coefficient equations imply $A (\hat{c} - c) = ρ$ , so $\hat{c} - c = A^{- 1} ρ$ . The endpoint formula is linear in the coefficient vector, hence $Q (\hat{c}) - Q = d^{⊤} A^{- 1} ρ$ .

2.1givenF4step 1.1

The hypotheses on $f, g, ψ_{k}$ make $r \in C^{s} [a, b]$ . Apply [F4] to $r$ to obtain the displayed exact expansion and bound its integral remainder by $(b - a) ∣ ω ∣^{- s} ∥ T_{g}^{s} r ∥_{\infty}$ .

2.2F1F5F8step 1.2algebra

With $x = A^{- 1} ρ$ , [F8] gives $d^{⊤} x = ⟨ x, \overline{d} ⟩$ and $∥ \overline{d} ∥_{2} = ∥ d ∥_{2}$ ; Cauchy--Schwarz [F5] therefore gives $∣ d^{⊤} A^{- 1} ρ ∣ \leq ∥ d ∥_{2} ∥ A^{- 1} ∥_{2} ∥ ρ ∥_{2}$ . The definition $κ_{2} (A) = ∥ A ∥_{2} ∥ A^{- 1} ∥_{2}$ and nonsingularity give the displayed condition-number form.

3.1givenF4F5F7step 2.1algebra

Since $c = A^{- 1} y$ , $∥ c ∥_{2} \leq ∥ A^{- 1} ∥_{2} ∥ y ∥_{2}$ . Linearity of $T_{g}$ gives $T_{g}^{j} r = T_{g}^{j} f - \sum_{k} c_{k} T_{g}^{j} (L_{ω} ψ_{k})$ . At each endpoint, [F5] bounds the corresponding divided sum by $∥ c ∥_{2} ∥ v_{j} (a) ∥_{2}$ or $∥ c ∥_{2} ∥ v_{j} (b) ∥_{2}$ ; [F7] combines it with the amplitude term. On the interval [F5] bounds $∥ \sum_{k} c_{k} T_{g}^{s} (L_{ω} ψ_{k}) ∥_{\infty}$ by $∥ c ∥_{2} K_{s}$ , and [F7] gives $∥ T_{g}^{s} r ∥_{\infty} \leq ∥ T_{g}^{s} f ∥_{\infty} + ∥ c ∥_{2} K_{s}$ . Substitution in the expansion from step 2.1 gives the stated $E_{j}, V_{j}, K_{s}$ bound.

4.1F7step 3.1step 2.2algebra∎

The complex triangle inequality [F7] combines the exact-collocation bound in step 3.1 with step 2.2, giving the stated integral bound for $Q (\hat{c})$ .

Remarks

The residual $ρ$ in the theorem is the exact residual of the stated matrix equation. If matrix products and solves are rounded, this theorem alone does not certify that exact residual or a rounded solve.

Depends on

Used by

A small Levin collocation residual can hide a large endpoint error Example

Dependency tree · two levels

43 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

D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, §1 (standard reference, not scraped)
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §3.3 (standard reference, not scraped)
R. M. Stuart and J. Voss, Matrix Analysis and Algorithms, §2 (standard reference, not scraped)