$\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{(-1/2k,1/2k)}$

Question

How do I calculate the following limit:

$$\lim_{k \rightarrow \infty} g_k(x) =\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{\left(\frac{-1}{2k},\frac{1}{2k}\right)}$$

Answer 1

To compute the limit, take a test function $\varphi$ and integrate:

$$\int_{-\frac{1}{2k}}^{\frac{1}{2k}} k(1+\cos (2\pi kx))\varphi(x)\,dx = k\int_{-\frac{1}{2k}}^{\frac{1}{2k}}\varphi(x)\,dx + \int_{-1/2}^{1/2}\cos (2\pi y)\varphi\left(\frac{y}{k}\right)\,dy.$$

The first integral tends to $\varphi(0)$, and the second to

$$\varphi(0)\int_{-1/2}^{1/2} \cos (2\pi y)\,dy = \varphi(0) \frac{1}{2\pi}\left(\sin \pi - \sin (-\pi)\right) = 0,$$

so $\lim\limits_{k\to\infty} g_k = \delta$.