Monday, 6 March 2017

real analysis - Show limarightarrowinftyint10f(x)xsin(ax2)=0.




Suppose f is integrable on (0,1), then show lim





I tried to write (0,1) = \bigcup _{k=0}^{{a-1}} \left(\sqrt{\frac{k}{a}},\sqrt{\frac{k+1}{a}}\right), but cannot make the integral converge to 0.


Answer



Suggestion:



Split [0, 1]
into the intervals where
ax^2 = 2\pi n,
ax^2 = \pi(2 n+1),
ax^2 = \pi(2 n+2).

These are
I_{2n} =[\sqrt{\frac{2\pi n}{a}}, \sqrt{\frac{\pi(2 n+1)}{a}})
and
I_{2n+1} =[\sqrt{\frac{\pi(2 n+1)}{a}}, \sqrt{\frac{\pi(2 n+2)}{a}}) .



Since

\sin(ax^2) > 0
in
I_{2n}
and
\sin(ax^2) < 0
in
I_{2n+1},
show that
the integral over
I_{2n} \cup I_{2n+1}

goes to zero.


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