I am reading a paper and the following came up:
Given a probability density function, $\rho(x)$, such that for $\epsilon > 0$ $ \int_{-\infty}^{\infty} |\rho(x)|^{1+\epsilon}dx < \infty $ $ \int_{-\infty}^{\infty} x^2 \rho(x) dx = \tau^2 < \infty $ and $ \rho(-x) = \rho(x) $ the following hold:
(i) $ \forall \mu_1 > 0$, $ \exists c_1 > 0$, possibly depending on $\mu_1$, s.t. when $|f| \ge \mu_1$ , we have $|\hat{\rho}(f)| < e^{-c_1} $
(ii) For any $n>n_0$, where $n_0$ is the solution to $\frac{1}{1 + \epsilon} + \frac{1}{n_0} = 1$, then $ \int_{-\infty}^{\infty} |\hat{\rho}(f)|^n \le \int_{-\infty}^{\infty} |\hat{\rho}(f)|^{n_0} = C_0 < \infty$
(iii) $\hat{\rho}(f) \to 0$ as $|f|\to \infty$
(iv) $ \exists c_2 > 0 $ s.t. for $\mu_1> 0$ small enough, whenever $|f| \le \mu_1$ we have $|\hat{\rho}(f)| \le e^{-c_2 f^2}$
I don't really know how to even approach these (standard? obvious?) properties. All the properties I know about (scaling, linearity, translation, etc.) don't seem to apply here. I got as far as the (trivial) step of noticing that the Fourier transform is real since the distribution is symmetric, but I couldn't make any further progress.
Any hints would be appreciated. More generally, references to literature that covers material useful for this type of analysis would be appreciated.
I am also confused as to the $\int_{-\infty}^{\infty}|\rho(x)|^{1+\epsilon}dx < \infty$ criterea...can anyone shed some intuition on the motivation for this?