Asymptotics of certain integrals

Question

For $n\in\mathbb N$ put $\displaystyle a_n=\int_0^{2\pi}\int_0^{2\pi}\frac{\cos2n(x-y)}{\sqrt{|x-y|}}dxdy$ and $b_n=\displaystyle \int_0^{2\pi}\int_0^{2\pi}\frac{\sin2n(x-y)}{\sqrt{|x-y|}}dxdy$. Can one give asymptotics for $a_n$ and $b_n$? In particular does there exist $0<p<1$ such that the series $$\sum_{n=1}^\infty(a_n^2+b_n^2)^{p}$$ is convergent?

Answer 1

For $a_n$,

$\displaystyle\int\int \frac{\cos(2n(x-y))}{\sqrt{|x-y|}} dxdy = \int_0^{2\pi}\left(\int_{r=-y}^{2\pi-y} \frac{\cos(2nr)}{\sqrt{|r|}}dr\right) dy \\ =\displaystyle \int_0^{2\pi}\left(\int_{r=0}^{y} + \int_{r=0}^{2\pi -y} \frac{\cos(2nr)}{\sqrt{r}}dr\right) dy$

inside integral,

$\int_0^x\frac{\cos(2nr)}{r^{1/2}}dr = \frac{C}{\sqrt{n}} \int_0^{2nx^2} \cos(t^2)dt$

thus the original integral is less than

$\frac{C}{\sqrt{n}}\sup_{0\le x\le 2\pi} \int_0^{2nx^2}\cos(t^2)dt$

it is about Fresnel integral. The integral is always finite. Thus

your integral is with decay rate as $\frac{1}{\sqrt{n}}$.

For the other part is the same way, $|b_n|\le C\frac{1}{\sqrt{n}}$, then

$\displaystyle\sum (a_n^2 + b_n^2)^p \sim C \sum \frac{1}{n^{p}}$

when $p<1$, it is not convergent.