Approaches to Solving Fredholm Integral Equations of the First Kind

Projection onto Kernel and Uniform Convergence

Given a Fredholm integral equation of the first kind:

$$ \int_a^b K(x, y) \phi(y) , dy = f(x), $$

where (K(x, y)) is the kernel, (\phi(y)) is the unknown function, and (f(x)) is a known function.

The approach involves projecting a set of functions onto the kernel (K(x, y)), calculating weights by the inner product:

$$ w_i = \langle \psi_i, K \rangle = \int_a^b \psi_i(y) K(x, y) , dy, $$

where ({\psi_i}) is a set of orthogonal polynomials converging point-wise to the reproducing kernel.

The goal is to achieve uniform convergence of the sequence of polynomials to the solution, represented as:

$$ \phi_n(x) = \sum_{i=1}^{n} w_i \psi_i(x), $$

ensuring that (\lim_{n \to \infty} \phi_n(x) = \phi(x)) uniformly.

Eigenfunction Derivation and Closed-Form Solutions

The eigenfunctions can be derived by substituting the basis into the integral equation:

$$ \int_a^b K(x, y) \left( \sum_{i=1}^{\infty} c_i \psi_i(y) \right) dy = f(x), $$

applying Fubini's theorem to swap summation and integration, leading to term-wise integration.

Connection to Variational and Galerkin's Methods

Variational Method

The variational method minimizes the functional:

$$ J[\phi] = \left| \int_a^b K(x, y) \phi(y) , dy - f(x) \right|^2, $$

where the solution (\phi) minimizes (J[\phi]) over an appropriate function space.

Galerkin's Method

Galerkin's method seeks to solve:

$$ \int_a^b K(x, y) \phi(y) , dy - f(x) = 0, $$

by ensuring the residual is orthogonal to the space spanned by the chosen basis ({\psi_i}):

$$ \langle \int_a^b K(x, \cdot) \phi(\cdot) , d\cdot - f(x), \psi_i(x) \rangle = 0, \quad \forall i. $$

Synthesis and Unified Approach

The approach combines the principles of the variational method and Galerkin's method, utilizing a point-wise converging set of polynomials, ensuring uniform convergence and deriving eigenfunctions through a sophisticated application of functional analysis and spectral theory.