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

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Repeated integration by parts for a nonstationary oscillatory integral

Statement

Let $a < b$ , $r \geq 1$ , $ω \in R ∖ {0}$ , $g \in C^{r + 1} [a, b]$ be real-valued with $∣ g^{'} ∣ \geq c > 0$ , and $h \in C^{r} [a, b]$ be complex-valued. With $T_{g} h = (h / g^{'})^{'}$ and $T_{g}^{0} h = h$ as in Finite-interval oscillatory integral and phase convention, one has $I_{ω} (h, g; [a, b]) = \sum_{j = 0}^{r - 1} \frac{(- 1)^{j}}{(i ω)^{j + 1}} {[\frac{T_{g}^{j} h (x)}{g^{'} (x)} e^{i ω g (x)}]}_{a}^{b} + \frac{(- 1)^{r}}{(i ω)^{r}} I_{ω} (T_{g}^{r} h, g; [a, b]) .$ Consequently, $∣ I_{ω} (h, g; [a, b]) - \sum_{j = 0}^{r - 1} \frac{(- 1)^{j}}{(i ω)^{j + 1}} {[\frac{T_{g}^{j} h (x)}{g^{'} (x)} e^{i ω g (x)}]}_{a}^{b} ∣ \leq \frac{b - a}{∣ ω ∣^{r}} ∥ T_{g}^{r} h ∥_{\infty} .$ For fixed $h, g, r$ , the remainder is $O (∣ ω ∣^{- r})$ ; the constant is the displayed derivative norm, not a frequency-uniform assertion about varying $h$ or $g$ .

Facts & Assumptions

Given: $a < b$ , the hypotheses in the Statement, and the componentwise complex Riemann integral from Finite-interval oscillatory integral and phase convention.

[F1]

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

[F2]

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 $u, v$ on a closed interval, $\int u v^{'} = [u v] - \int u^{'} v$ . Applying this to real and imaginary components gives the componentwise complex identity.

[F3]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : sums, products, and quotients with nonzero denominator preserve differentiability and obey the stated derivative rules.

[F4]

For continuous complex $z$ on $[a, b]$ , the componentwise definition of Finite-interval oscillatory integral and phase convention and the conjugation and modulus laws Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive give $∣ \int_{a}^{b} z ∣ \leq \int_{a}^{b} ∣ z ∣$ . If $Z = \int z \neq 0$ , set $ζ = \overline{Z} / ∣ Z ∣$ , so $∣ ζ ∣ = 1$ and $∣ Z ∣ = \int Re (ζ z)$ . Since $Re (ζ z) \leq ∣ z ∣$ , monotonicity from Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in $R^{m}$ proves the bound; the zero case is immediate.

Proof

1.1givenF1F2F3algebra

Since $g^{'}$ is continuous and nowhere zero, [F3] makes $h / g^{'}$ differentiable with derivative $T_{g} h$ . Apply [F2] componentwise with $u = h / g^{'}$ and $v = e^{i ω g}$ . By [F1], $v^{'} = i ω g^{'} v$ , so rearranging gives $I_{ω} (h, g) = [(h / g^{'}) v]_{a}^{b} / (i ω) - I_{ω} (T_{g} h, g) / (i ω)$ .

2.1givenF1F4step 1.1algebra∎

Apply step 1.1 successively to $h, T_{g} h, \dots, T_{g}^{r - 1} h$ . At each application the final integral receives one further factor $- 1 / (i ω)$ and the next endpoint term receives the matching alternating sign, yielding the displayed finite sum and remainder. Since $∣ e^{i ω g (x)} ∣ = 1$ , [F4] bounds the remainder modulus by $∣ ω ∣^{- r} \int_{a}^{b} ∣ T_{g}^{r} h (x) ∣ d x \leq (b - a) ∣ ω ∣^{- r} ∥ T_{g}^{r} h ∥_{\infty}$ .

Depends on

Used by

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Sources

NIST Digital Library of Mathematical Functions, §2.3(i) (standard reference, not scraped)
D. Levin, Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations, §3 (standard reference, not scraped)
A. Iserles, S. P. Nørsett, and S. Olver, Highly oscillatory quadrature: The story so far, §2 (standard reference, not scraped)