Toronto Math Forum

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

Title: TT2--Problem 3
Post 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
Title: Re: TT2--Problem 3
Post by: Ian Kivlichan on November 15, 2012, 10:30:00 PM
Hopeful solution attached! :)

EDIT: was not originally attached..?
Title: Re: TT2--Problem 3
Post by: Victor Ivrii on November 16, 2012, 06:59:23 AM
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.