I've got two exercises about sequences of functions:
- If $a < b$, find a function $\varphi \in \mathcal{C}^{\infty}$ that is null outside $(a, b)$, and it must be positive in that interval and $$\int\limits_{-\infty}^{\infty} \varphi(x)dx = 1.$$
Let $f(x;a)=\exp(-\frac{1}{x-a}) \chi_{(a, \infty)}, g(x;b)=\exp(-\frac{1}{b-x})\chi_{(-\infty,b)}, h(x) = f \cdot g, \lambda = \frac{1}{\int_{-\infty}^{\infty}h(x)dx}$ and $\varphi(x)= \lambda h(x)$.
The second exercise asks me:
- Let $a = -\frac{1}{n}$ and $b = \frac{1}{n}$ in the exercise above and specify the function with $\varphi_{n}$, prove that if $f$ is uniformly continuous $\forall x \in \mathbb{R}$, then $$f * \varphi_{n} \rightrightarrows f,$$ where $f*g(x) = \int\limits_{-\infty}^{\infty} f(y)g(x-y)dy =\int\limits_{-\infty}^{\infty} f(x-y)g(y)dy. $
But I haven't got any clue on how to proceed. I'm quite desperate, so whichever help you can give me is appreciated, preferrably more than less.
Thank you beforehand.
Edit: Found a $\mathcal{C}^{\infty}$ function that complies.