TT2--Problem 3

[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.