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

LetThe integral is absolutely convergent. For a Schwartz function $\phi$, Fubini's theorem and integration by parts in $\lambda$ giveEquivalently, $v^{\prime }=-\hat{u}$ as a distributional derivative, and hence

Because $x^{-2}$ is integrable on $[1,\infty )$, the Dominated convergence theorem applied to the defining integral proves that $v$ is continuous on $\mathbb{R}$.

For $\lambda >0$, the change of variables formula $t=\lambda x$ givesand for $\lambda <0$ the analogous formula is obtained with $e^{it}$. On either open half-line the lower endpoint stays away from zero locally, so repeated differentiation under the integral sign proves smoothness. Thus

For $0<\lambda <1$, the preceding change of variables and one integration by parts giveSplit the last improper integral at one and add and subtract one on $(\lambda ,1)$. Since ${\int }_{\lambda }^{1}dx/x=-\log \lambda$,

The definition also gives the complex conjugate relationConsequently, for $-1<\lambda <0$,

On either open half-line, part (i) says $\hat{u}=-v^{\prime }$. Differentiating the formula in part (iii) makes all elementary terms cancel except the logarithm and the bracket. ThereforewhereandwhereBoth functions are smooth on their respective half-lines, proving the required assertion.

For $\lambda >0$, subtracting the two formulas from part (iv) yieldsUsing the stated Dirichlet integral and taking the one-sided limit gives

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.