Distributional limit of $m\sin(m|x^2+y^2-1|)$

Question

Consider the sequence $$ f_m(x,y)=m\sin(m|x^2+y^2-1|) $$ as sequence of distributions in $\mathcal{D}^\prime(\mathbb{R}^2\setminus\{0\})$. What is the limit as $m\to\infty$? In other words, given a smooth function $\varphi$ with compact support contained in $\mathbb{R}^2\setminus\{0\}$, what is $$ \lim_{m\to\infty}\int f_m(x,y)\varphi(x,y)\,dxdy? $$

Answer 1

Note that by an easy substitution \begin{align} \int_0^\infty\int_0^{2\pi} rm\sin(m|1-r^2|)\varphi_r(r,\phi)\,d\phi\,dr&=\frac12\int_0^1m\sin(mx)\varphi_r(\sqrt{1-x})\,dx\\&\phantom{=}+\frac12\int_0^\infty m\sin(mx)\varphi_r(\sqrt{1+x})\,dx. \end{align} Here, we set $\int_0^{2\pi}\varphi(r,\phi)\,d\phi=\varphi_r(r)$. Using integration by parts in both integrals, it is easy to see that the right-hand side converges to $\varphi_r(1)$. In summary, $f_m\to f$ in $\mathcal{D}^\prime(\mathbb{R}^2\setminus\{0\})$ where $$ \langle f,\varphi\rangle=\int_{0}^{2\pi}\varphi(1,\phi)\,d\phi. $$