GalerkinsMethodForCovarianceOperatorEigenfunctions

See also: FredholmIntegralEquationToSturmLiouvilleform

Goal

Determine the eigenfunctions of an integral covariance operator with kernels belonging to the class of absolutely continuous, positive definite, infinitely differentiable functions on $\mathbb{R}$, by transforming a pointwise convergent polynomial basis into a uniformly convergent series and subsequently into a Sturm-Liouville problem.

Background

• Kernel Class: The focus is on translation-invariant kernels $K(h)$ where $h = |t - s|$] and $K$ belongs to the class of absolutely continuous, positive definite, infinitely differentiable functions on $\mathbb{R}$.
• Mercer's Theorem: Ensures the existence of a uniformly convergent series for such kernels.

Methodology

1. Galerkin Method for Uniform Convergence

• Projection: Apply Galerkins method to project the kernel $K(h)$ onto a polynomial basis:

To properly identify $c_m$ in the equation, let's follow these steps:

1. Start with the integral equation:

$$ \phi(x) = \lambda \int_{a}^{b} K(x, y) \phi(y) dy $$

2. Expand $\phi(x)$ using the orthogonal basis $P_n(x)$:

$$ \phi(x) = \sum_{n=0}^{\infty} c_n P_n(x) $$

3. Substitute this into the integral equation:

$$ \sum_{n=0}^{\infty} c_n P_n(x) = \lambda \int_{a}^{b} K(x, y) \sum_{n=0}^{\infty} c_n P_n(y) dy $$

4. Multiply both sides by $P_m(x)$ and integrate over $[a, b]$:

$$ \int_{a}^{b} P_m(x) \sum_{n=0}^{\infty} c_n P_n(x) dx = \lambda \int_{a}^{b} P_m(x) \int_{a}^{b} K(x, y) \sum_{n=0}^{\infty} c_n P_n(y) dy dx $$

5. Using the orthogonality of $P_n(x)$:

$$ c_m = \lambda \int_{a}^{b} \int_{a}^{b} K(x, y) P_m(y) P_m(x) dy dx $$

This isolates $c_m$ on one side of the equation, providing a formula to compute each coefficient $c_m$ in the expansion of $\phi(x)$.

This process transitions from a polynomial basis for the kernel that converges pointwisely to uniform convergence due to the conditions for Mercer's theorem being satisifed. See PointwiseVsUniformConvergence

2. Series Expansion and Fubini's Theorem

• Series Substitution: Replace the kernel $K(h)$ in the integral eigenvalue equation with its uniformly convergent series expansion:

$$ \sum_{n=0}^{\infty} c_n P_n(x) \left( \int_{a}^{b} P_n(y) \phi(y) dy \right) = \lambda \phi(x) $$

• Applying Fubini's Theorem: The absolute continuity and infinite differentiability of $K(h)$ allow for the interchange of summation and integration:

$$ \int_{a}^{b} \left( \sum_{n=0}^{\infty} c_n P_n(x) P_n(y) \right) \phi(y) dy = \lambda \phi(x) $$

3. Sturm-Liouville Differential Equation Formation

• Differentiation: Differentiate the modified equation twice with respect to $x$ to form a Sturm-Liouville differential equation.

4. Eigenfunction Determination

• Solving the Equation: Solve the Sturm-Liouville differential equation to determine the eigenfunctions $\phi(x)$ for each term.

This methodology provides a rigorous approach for finding eigenfunctions of integral covariance operators with kernels from this specific class, utilizing advanced mathematical techniques and transformations.