Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 1 / b / vi

Remove the two reciprocal tails by settingThe assumed $O(x^{-2})$ remainder at each end and local integrability on bounded intervals imply $r\in L^1(\mathbb{R})$. Its Fourier transform $\hat{r}$ is therefore continuous at zero. Since $\widehat{u(-\cdot )}(\lambda )=\hat{u}(-\lambda )$,Parts (iv) and (v) show that both requested one-sided limits exist. If $L_{+}$ denotes the limit from positive frequencies and $L_{-}$ the limit from negative frequencies, then the common value $\hat{r}(0)$ cancels andThis is the universal jump caused by reciprocal tails.