Author Topic: TT2--Problem 3  (Read 4585 times)

Victor Ivrii

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TT2--Problem 3
« 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

Ian Kivlichan

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Re: TT2--Problem 3
« Reply #1 on: November 15, 2012, 10:30:00 PM »
Hopeful solution attached! :)

EDIT: was not originally attached..?
« Last Edit: November 15, 2012, 10:32:49 PM by Ian Kivlichan »

Victor Ivrii

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Re: TT2--Problem 3
« Reply #2 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) -- 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.