J0CovarianceIntegralOperator

Covariance Kernel and Integral Operator

Given the covariance kernel:

$$K(x, y) = \pi J_0(2\pi |x-y|)$$

the associated integral operator $A$ acting on a function $f$ is:

$$(Af)(x) = \int_\Omega \pi J_0(2\pi |x-y|) f(y) dy$$

Polynomial Orthonormal Basis and Eigenvalue Problem

Assuming we have a polynomial orthonormal basis ${ p_n(x) }$ which is orthonormal with respect to the weight function $w(x) = 1$, to determine if these polynomials are eigenfunctions of $A$ and to find the corresponding eigenvalues, we must solve the following integral equation for each $n$:

$$\int_\Omega \pi J_0(2\pi |x-y|) p_n(y) dy = \lambda_n p_n(x)$$

If the polynomial $p_n(x)$ satisfies the above equation for a scalar $\lambda_n$, then $p_n(x)$ is an eigenfunction of $A$ with $\lambda_n$ as its corresponding eigenvalue.

Karhunen-Loève Expansion

Given the eigenfunctions $p_n(x)$ and eigenvalues $\lambda_n$ of the operator $A$, the KL expansion of the GP $X(t)$ is:

$$X(t) = \sum_{n=1}^{\infty} Z_n \sqrt{\lambda_n} p_n(t)$$

where $Z_n$ are independent standard Gaussian random variables.