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

Jacob, Niels; Evans, Kristian P

ISBN 10: 9813274522 / ISBN 13: 9789813274525

Editore: WSPC, 2018

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Dati bibliografici

• Titolo: Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations
• Editore: WSPC
• Data di pubblicazione: 2018
• Lingua: Inglese
• ISBN 10: 9813274522
• ISBN 13: 9789813274525
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

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.

Recensione

Like its predecessors in this series, the book is an excellent reference for anyone interested in these topics. The authors did not lower the standards with respect to both clarity of the presentation and depth of the material. The reader can sense everywhere in the book the rich experience of the authors in teaching Mathematics, in particular Analysis in the broader sense, and the best proof for this are the fine balance they found between the hard theoretical results and applications, on one hand, and the appropriate examples that illustrate the theoretical results. --zbMATH