I've been working on the Fundamental solution of Homogenous Heat Equation and I have problem with the following equality.
$$\lim_{\epsilon \to 0} \int_{\mathbb{R}} \frac{1}{2\sqrt{\pi}} \exp\left(\frac{-y^2}{4}\right)\phi\left(\sqrt{\epsilon}y, \epsilon\right)dy = \phi(0,0) \int_{\mathbb{R}} \frac{1}{2\sqrt{\pi}} \exp\left(\frac{-y^2}{4}\right)dy.$$
How can I proof the following equality?
$$ \lim_{\epsilon \to 0} \int_{\mathbb{R}} \frac{1}{2\sqrt{\pi}} \exp\left(\frac{-y^2}{4}\right)\phi\left(\sqrt{\epsilon}y, \epsilon\right)dy = \int_{\mathbb{R}} \lim_{\epsilon \to 0}\left(\frac{1}{2\sqrt{\pi}} \exp\left(\frac{-y^2}{4}\right) \phi\left(\sqrt{\epsilon}y, \epsilon\right)\right)dy.$$
I want to know how I can formally proof that the limit goes inside the Lebesgue integral.
Notes:
- Suppose that $\phi$ is a test function.
- The integral is a Lebesgue integral.
- $\delta$ is Dirac Delta Distribution. ($\delta(\phi) = \phi(0), \phi \in D(\Omega)$ where $D$ is the space of Test Functions).
- $E$ is a fundamental solution of the differential operator with constant coefficients $P$ if $P(E) = \delta$.