Why does bessel's method work?

Bessel's method works because it provides a systematic approach to approximate functions using orthogonal polynomials, specifically Bessel functions, which are solutions to Bessel's differential equation. These functions are particularly useful in problems involving cylindrical or spherical symmetry, allowing for effective representation and manipulation of complex functions. The orthogonality property of Bessel functions ensures that the coefficients in the expansion can be accurately determined, leading to better numerical approximations and solutions in various applications, such as signal processing and physics.