316 Math (Sturm-Liouville theory)

1. Inner product space.

Vector space; inner product space; the space of square integrable functions L²; convergence in L²; orthogonal functions (9 lectures).

2. The Sturm-Liouville theory.

Linear second-order equations; self-adjoint differential operator; the Sturm-Liouville problem; existence and completeness of the eigenfunctions; regular and singular problems (9 lectures).

3. Fourier series.

The fundamental theorem of Fourier series in L²; pointwise theory of Fourier series; applications to boundary-value problems (6 lectures).

4. Orthogonal polynomials.

Legendre, Hermite, and Laguerre polynomials as solutions of certain singular Sturm-Liouville problems; their orthogonality and completeness properties; generalized Fourier expansions (9 lectures).

5. Bessel functions.

The gamma function; Bessel's equation and Bessel functions of the first kind; orthogonality properties (6 lectures).

6. Fourier transformation. The fourier transform; the Fourier integral; properties and applications in PDE (6 lectures).