Hi just need a bit of help with a few parts of this practice question:
Show $F: (0, \infty) \rightarrow \mathbb{R}$ diff'ble with respect to $t \in (0, \infty)$, where
$$F(t) := \int_{\mathbb{R}}e^{-tx^2}\cos(x)dx$$
According to theorems in my text, I know I need to show 3 things:
- $f(x,t) = e^{-tx^2}\cos(x)$ is $\mu$ integrable. I.e. $\int_{\mathbb{R}}|e^{-tx^2}\cos(x)|dx < \infty$
I do this by showing that $\int_{\mathbb{R}}|e^{-tx^2}\cos(x)|dx < \int_{\mathbb{R}}|e^{-x^2}|dx < \infty$, by direct integration.
- I need to show that $\frac{\partial f}{\partial t}(x,t) = -x^2\cos(x)e^{-tx^2} $ exists for almost all of $x$ and is continuous on $(0, \infty)$.
This certainly seems continuous for $t \in (0, \infty)$ and seems to exist for all of $x$, so I am a bit confused about how to show that this exists for 'almost' all of $x$.
- Finally, I need to find a dominating function $h \in \mathcal{L}^1(x)$ such that $|\frac{\partial f}{\partial t}(x,t)| \le h(x)$
But I'm not quite sure how to construct $h(x)$ independent of $t$ to achieve this final requirement.
Any insight would be greatly appreciated!
There is no such function, as you can see for $t \to 0$.
But the point is that differentiability is a local property. Hence, it suffices for each interval $(t_0,t_1)$ with $0<t_0<t_1$ to find a dominating function which is independent of $t$ (but only needs to hold for $t$ in the above interval).
This is easy to do, namely $x^2 e^{-t_0 x^2}$ will do.