**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).

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