In the part on Fourier analysis, we discuss pointwise convergence
results, summability methods and, of course, convergence in the
quadratic mean of Fourier series. More advanced topics include a first
discussion of Hardy spaces. We also spend some time handling general
orthogonal series expansions, in particular, related to orthogonal
polynomials. Then we switch to the Fourier integral, i.e. the Fourier
transform in Schwartz space, as well as in some Lebesgue spaces or of
measures.Our treatment of ordinary differential equations starts with a
discussion of some classical methods to obtain explicit integrals,
followed by the existence theorems of Picard-Lindelöf and Peano which
are proved by fixed point arguments. Linear systems are treated in great
detail and we start a first discussion on boundary value problems. In
particular, we look at Sturm-Liouville problems and orthogonal
expansions. We also handle the hypergeometric differential equations
(using complex methods) and their relations to special functions in
mathematical physics. Some qualitative aspects are treated too, e.g.
stability results (Ljapunov functions), phase diagrams, or flows.Our
introduction to the calculus of variations includes a discussion of the
Euler-Lagrange equations, the Legendre theory of necessary and
sufficient conditions, and aspects of the Hamilton-Jacobi theory.
Related first order partial differential equations are treated in more
detail.The text serves as a companion to lecture courses, and it is also
suitable for self-study. The text is complemented by ca. 260 problems
with detailed solutions.
