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Let $F(x),G(x)$ be nonnegative decreasing functions in $[0,+\infty)$, with$\,\displaystyle \lim_{x\rightarrow+\infty}{x(F(x)+G(x))}=0$

(1) Prove that: $\forall \varepsilon>0$,we have $\displaystyle \lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}}=0$
(2) If we have $$\lim_{n\rightarrow+\infty}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{\frac{t}{n}} dt}}=0 $$ then prove that $$ \lim_{x\rightarrow0}{\int_{0}^{+\infty}{(F(t)-G(t))\cos{(xt)}dt} }=0 $$

I tried let $$ f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}} $$,then for a fixed value of x,by Dirichlet test,we can see the $$f(x)=\lim_{x\rightarrow+\infty}{\int_{\varepsilon}^{+\infty}{xF(xt)\cos{t}dt}} $$ is convergence,then I have no idea about the next step.:(

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    $\begingroup$ It would be much helpful if you demonstrate what you have tried. $\endgroup$ – Sangchul Lee Mar 10 '13 at 13:51
  • $\begingroup$ @sos440, I tried to prove the limit is onvergence,then I stucked here $\endgroup$ – pxchg1200 Mar 10 '13 at 14:51
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For (1) define a sequence by letting $$ a_k = \int_{\pi/2+k\pi}^{\pi/2+(k+1)\pi} x F(xt)\cos t\, dt $$ and notice that your integral is given by $$ \int_\varepsilon^{\pi/2} - \sum_{k=0}^\infty a_k. $$ Then notice that $a_k=(-1)^k |a_k|$ with $|a_k| \to 0$ and $|a_k|$ decreasing. So you can apply the alternating series convergence test to find convergence. Moreover you know that the limit of the series is comprised between the first two terms of the serie. Show that for $x\to \infty$ every term of the series goes to zero.

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  • $\begingroup$ I got it,how about (2) ? $\endgroup$ – pxchg1200 Mar 11 '13 at 12:11

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