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Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations (Hardcover)
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Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations (Hardcover)
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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-Lindeloef 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.
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