Golden-Window Fourier Diffraction

Abstract

The Fourier amplitude of a finite interval window is an exact sine kernel, specializing at golden-window length to the diffraction closed form.

Theorem 1.1 (A finite interval window has exact sine-kernel amplitude).

$$|{\hat{c}}_m(\ell )|=\frac{|\sin (\pi m\ell )|}{\pi m},m>0.$$

Proof. Machine-checked in Lean as D5/S3/Fourier/WindowDiffraction.window_fourier_amplitude (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a positive Fourier mode, integrating the complex exponential over the interval from zero to the window length and taking its norm gives the exact sine-kernel amplitude. The proof evaluates the exponential integral, reduces the complex norm to the sine half-angle identity, and uses positivity of the mode and pi to normalize the denominator.

Theorem 1.2 (The golden window has the diffraction closed form).

$$|{\hat{c}}_m|=\frac{|\sin (\pi m/\phi )|}{\pi m},\phi =\frac{1+\sqrt{5}}{2},m>0.$$

Proof. Machine-checked in Lean as D5/S3/Fourier/WindowDiffraction.golden_window_fourier_amplitude (✓ std3). ∎

Source. Repository-derived.

Commentary.

The cut-and-project interval window has length one over the golden ratio. Substituting this length into the general interval-window formula gives the exact diffraction amplitude |c-hat_m| = |sin(pim/phi)|/(pim), with no asymptotic approximation or omitted normalization factor.

References

• Truth anchor: D5/S3/Fourier/WindowDiffraction.golden_window_fourier_amplitude
• Truth anchor: D5/S3/Fourier/WindowDiffraction.window_fourier_amplitude