SpectralCovarianceMeasures

The Spectral Covariance Measure of $\frac{1}{\sqrt{1-t^2}}$

The covariance function $r(t) = \pi J_0(2\pi t)$ where $J_0$ is a Bessel function of the first kind of order 0 can be expressed in terms of the spectral density $\dot{\rho}(s)$ or spectral measure $\rho(s)$ as:

$$r(t) = \pi J_0(2\pi t) = \int_{-1}^{1} e^{2\pi i t s} \dot{\rho}(s) ds = \int_{-1}^{1} e^{2\pi i t s} d\rho(s)$$

where $\dot{\rho}(s) = \frac{1}{\sqrt{1 - s^2}} \forall s \in [-1, 1]$ and $d\rho(s) = \dot{\rho}(s) ds$.

The inverse Fourier transform of $\pi J_0(2\pi s)$ is the spectral density:

$$\dot{\rho}(s)=\frac{1}{\sqrt{1 - t^2}} = \int_{-\infty}^{\infty} r(s) e^{-2\pi i t s} ds = \int_{-\infty}^{\infty} \pi J_0(2\pi s) e^{-2\pi i t s} ds$$

whose integral is the spectral measure

$$\rho (t) = \int_{- 1}^t \dot{\rho} (s) ds = \int_{- 1}^t\frac{1}{\sqrt{1 - s^2}} ds = \arcsin (t) + \frac{\pi}{2}$$