APM346-2012 > Term Test 2

TT2--Problem 3

(1/1)

Victor Ivrii:
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:
Hopeful solution attached! :)

EDIT: was not originally attached..?

Victor Ivrii:
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.

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