Toronto Math Forum
APM3462012 => APM346 Math => Term Test 2 => Topic started by: Victor Ivrii on November 15, 2012, 08:22:48 PM

Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following conditions:
 $\phi$ is continuous.
 $\phi'$ is continuous.
 $\phi(x) = 0$ for all $x>1$.
Consider the integral
$$
I_\lambda = \int_{\infty}^{+\infty} \phi (x) \cos (\lambda x)\, dx.
$$
Prove that $I_\lambda \rightarrow 0$ when $\lambda \rightarrow \infty$.
Post after 22:30

Hopeful solution attached! :)
EDIT: was not originally attached..?

Actually since we integrate from $\infty$ to $\infty$ and $\phi$ has a bounded support (see my notes to Problem 1 (http://forum.math.toronto.edu/index.php?topic=137.msg776#msg776))  or in general fast decays  we can integrate by parts as many times as smoothness of $\phi$ allows getting $I_\lambda=O(\lambda^{s}$ where $s$ is the smoothness.