Past exam of the fourier analysis course of the University of Oxford / 2023 / 1 / b / ii / Solution

If $\alpha >1$, then $f_{\alpha }$ is an integrable function, so the Riemann-Lebesgue lemma supplies a continuous representative of its Fourier transform. Equivalently, in the mixture for $h_{\alpha }$ the majorant $t^{(\alpha -3)/2}{e}^{-t}$ is integrable at zero precisely when $\alpha >1$. In this case dominated convergence also givesIf $0<\alpha \le 1$, Fatou's lemma applied along any sequence of nonzero $\xi$ tending to zero givesThe divergence comes from zero, where the exponent is at most $-1$. A continuous representative $H$ would agree with $h_{\alpha }$ almost everywhere. Since both are continuous on each open half-line, they must agree everywhere on those half-lines: a nonzero continuous difference would be nonzero on an interval of positive Lebesgue measure. The displayed blow-up then contradicts continuity of $H$ at zero. Changing a value at one point cannot remove this singularity. Consequently