Fourier series, Fourier transform and their applications to mathematical physics
This book is a concatenation of four essentially self-contained parts, each of ca. 130 pages and based on half-semester graduate courses given by the author at the University of Oulu. These parts are: (I) Fourier series and the discrete Fourier transform, (II) Fourier transform and distributions, (III) Operator theory and integral equations, and (IV) Partial differential equations. The material is rather extensive, and both its content and order are sometimes unexpected. After, and also between, the expected standard topics, it goes quite far in some selected directions. Part I develops the theory of periodic Hölder and Besov spaces, and proves the absolute convergence of Fourier series in such spaces, before discussing Dirichlet's kernel and Dini's condition for pointwise convergence. Parts II--IV have lengthy and technical discussion of the Helmholtz, Schrödinger and magnetic Schrödinger operators occupying several chapters. The usual heat and wave operators are left at the very end of Part IV after much more delicate results on scattering problems. The 46 chapters are quite uneven in nature. Some of them are short and easily digestible, perfect for a student's self-study or for a teacher to convert into a beautiful lecture with minimal preparation. Others, like the technical 30-page Chapter 23, seem to serve a very different purpose and audience. Overall, the style is somewhat intermediate between ``mathematics proper'' and ``mathematical methods''. There are some minor oversights in mathematical rigour, like in the proof of Theorem 18.18, where convergence in the Schwartz class is claimed, although only pointwise convergence is checked. The origin of the text as four sets of lecture notes is at times too visible in that it often lacks the usual finishing touches with informal motivation and discussion, but instead some chapters consist of mere definitions, propositions and proofs, one after the other. Sometimes only by reading a proof one discovers that it was not the expected complete argument, like in the case of the spectral theorem where, after two pages of reduction to the bounded case, it is revealed that ``for bounded operators we will only sketch the proof''. At other places, there are rather advanced implicit prerequisites, like the casual mention (p. 48) that ``for noninteger \(\alpha\) one needs to interpolate between the spaces \(C^{[\alpha]}\) and \(C^{[\alpha]+1}\)''. Zygmund is consistently misspelled as ``Zigmund''.
scientific article; zbMATH DE number 6764600