What a sharp rectangular pulse costs in wavenumber space

Use the Fourier-transform convention \[ F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}\,dx, \qquad f(x)=\frac1{2\pi}\int_{-\infty}^{\infty}F(k)e^{ikx}\,dk. \] Let \(f(x)=A\) for \(|x|<a\) and \(f(x)=0\) for \(|x|>a\), where \(A\) is real and nonzero. Values at \(x=\pm a\) may be chosen arbitrarily. Derive \(F(k)\), including its continuous value at \(k=0\), and locate its first zeros and central-lobe width. Verify Parseval's identity for this pair and thereby derive \[ \int_{-\infty}^{\infty}\left(\frac{\sin u}{u}\right)^2du=\pi. \] Then determine whether the spectral second moment \(\int k^2|F(k)|^2dk\) is finite. Explain precisely why compact support does not imply finite root-mean-square bandwidth here, how the jump discontinuities control the high-\(|k|\) tail, and why changing the pulse's two endpoint values does not change its transform as an \(L^1\) function.

Integrate \(Ae^{-ikx}\) from \(-a\) to \(a\), then take the \(k\to0\) limit rather than substituting into a quotient.