ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Transform of an interval indicator

Example

Assume countable choice and $a \leq b$ . For $f = 1_{[a, b]}$ on $R$ , $\hat{f} (ξ) = \frac{e^{- 2 π i a ξ} - e^{- 2 π i b ξ}}{2 π i ξ} (ξ \neq 0), \hat{f} (0) = b - a .$ In particular for $[a, b] = [- 1 / 2, 1 / 2]$ the transform is $\sin (π ξ) / (π ξ)$ with value one at zero.

Facts & Assumptions

Given: $a \leq b$ and The Axiom of Countable Choice ( ${A C}_{ω}$ ), with the integral convention of Fourier transform on complex L1 classes.

[F1]

Complex FTC evaluates continuous derivatives on intervals (Complex integration by parts on intervals and decaying lines).

[F2]

Euler's formula and the real sine/cosine derivatives give the exponential derivative ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ , The derivatives of sine and cosine are cosine and minus sine).

[F3]

The transform of an integrable function is continuous (The L1 transform is bounded and uniformly continuous).

Verification

1.1F1F2given

The interval indicator is integrable with norm $b - a$ . For $a < b$ and $ξ \neq 0$ , F2 gives the antiderivative $- e^{- 2 π i x ξ} / (2 π i ξ)$ . F1 at a and b gives the displayed quotient. If $a = b$ the indicator is null almost everywhere and both the numerator and transform vanish.

2.1F1F2F3step 1.1∎

At zero frequency the integral is the interval length $b - a$ ; F3 shows this is the continuous extension of the quotient. For the symmetric interval its numerator is $e^{π i ξ} - e^{- π i ξ} = 2 i \sin (π ξ)$ by F2, giving the sinc formula and value one. Countable choice is inherited from the interval Lebesgue measure and FTC bridge.

Depends on

Used by

An L1 transform need not be integrable Counterexample
Triangle function and squared sinc Example

Dependency tree · two levels

38 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

Gerald Teschl, Topics in Real and Functional Analysis (2017) (standard reference, not scraped)