Fourier inversion for $\hat\varphi^2$: $\int \hat\varphi(r)^2 \cos(ry)\,dr = 2\pi A_\varphi(y)$

Throughout, $\varrho$ is a taper profile (TaperProfile): a monotone $C^3$ function $\mathbb{R}\to\mathbb{R}$ equal to $0$ on $(-\infty,0]$ and $1$ on $[1,\infty)$. For a window length $L$ and ramp width $w$, the taper is $\varphi(u) = \varrho\bigl((L/2 - |u|)/w\bigr)$: an even function with $0\le\varphi\le 1$, supported in $[-L/2,L/2]$ and equal to $1$ on $[-L/2+w,\,L/2-w]$. The paper's Fourier transform is $h_f(z) = \int_{\mathbb{R}} f(u)\,e^{izu}\,du$ (paperFT), and $\hat\varphi(r)$ denotes the real part of $h_\varphi$ at a real argument $r$ (phiHatR).

Let $A_\varphi = \varphi \star \varphi$ be the autocorrelation $A_\varphi(y) = \int \varphi(u)\,\varphi(u+y)\,du$, as in [eq:PhigA].

Assume $w > 0$ and $2w \le L$. Then for every real $y$,

Since $\hat\varphi^2 = (A_\varphi)^{\wedge}$ in the paper's convention, this is Fourier inversion in cosine form; it is the identity used in the P-part of [prop:trace] ($\int \hat\varphi(\tau - \tau_k)^2 \cos(\tau y)\,d\tau = 2\pi A_\varphi(y)\cos(\tau_k y)$). It is consumed by Zeta23.PrimeSide.localHyps_concrete and, at $y = 0$, yields the Plancherel identity Zeta23.Taper.integral_phiHatR_sq.