Sturm-Liouville Form of the Integral Covariance Operator with Bessel Function Kernel J₀

Objective

Detail the process of determining eigenfunctions for an integral covariance operator with the Bessel function kernel $J_0$, formulating it as a Sturm-Liouville problem.

Background and Relevance of Sturm-Liouville Problem

• Sturm-Liouville Problem: Essential for solving a range of physical and mathematical problems, particularly for representing eigenvalue problems and determining corresponding eigenfunctions.

Methodology

1. Galerkin Method for Uniform Convergence

• Application: The Galerkin method projects a function onto a subspace of trial functions, transforming a pointwise convergent series into a uniformly convergent one.
• Projection Process:

$$c_n = \frac{\int J_0(x) P_n(x) dx}{\int P_n^2(x) dx}$$

2. Uniform Convergence and Fubini's Theorem

• Fubini's Theorem Application: The theorem's conditions — absolute continuity and infinite differentiability of the kernel $J_0$ — allow for the interchange of summation and integration, essential for formulating the Sturm-Liouville problem.
• Interchange Process: From:

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

To:

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

3. Sturm-Liouville Formulation

• Differentiation Process: Transforming the modified equation into a Sturm-Liouville differential equation is achieved through differentiation, setting the stage for eigenfunction determination.

4. Eigenfunction Determination

• Solving Techniques: Employ methods like separation of variables, power series, or numerical approaches to solve the Sturm-Liouville equations and find the eigenfunctions.

Conclusion

This methodology ensures a comprehensive and precise approach to finding eigenfunctions for the integral covariance operator with the Bessel function kernel $J_0$, utilizing advanced mathematical principles and techniques.