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Fourier Series, Fourier Transforms, and Function Spaces: A Second Course in Analysis

Tim Hsu
Publication Date: 
Number of Pages: 
AMS/MAA Textbooks
[Reviewed by
Allen Stenger
, on
This is an interesting take on the second course in analysis: rather than the Lebesgue integral, we study Fourier analysis and applications. The book is well done and makes a strong case for this approach. The Introduction (which is the Introduction for the Instructor) is one of the best I’ve read, and you should definitely study if you are considering adopting the book. It explains very clearly the goals of the book, the limitations of this approach, and some other unusual features of the book.
Today we think of the Lebesgue integral as being essential to the study of Fourier analysis, at least for mathematicians (engineers and physicists are skilled at Fourier analysis but are usually oblivious to the Lebesgue integral). Historically Fourier’s work preceded both the Riemann and Lebesgue integrals and preceded Riemann and Lebesgue themselves.  This book is unusual in that some of the theorems are proved in the body of the book, but most are relegated to the problems. The theorems are broken up into small enough pieces that none of the pieces are very difficult to prove, and there are also one-line hints for most of the problems in an appendix. For example, the theorem that the Cesàro sums of the Fourier series of a continuous function converge uniformly to the function is broken into four problems, two about properties of the Fejér kernel and two about estimating the errors in the sums. This is in addition to a theorem, whose proof is in the body, giving an explicit form for the Fejér kernel.  The problems also include a number of theorems and proofs that are not in the main path (these are marked with asterisks, and do not have hints 1in the back), and a few non-proof problems such as evaluating the Riemann zeta functions \( \zeta(2) \) and \( \zeta(4) \).  
The book is divided into four parts. Part 1, which is the first third of the book, reviews what should have been in the first course in analysis and reworks it to also handle complex-valued functions. It starts with an axiomatic definition of the real numbers (and so does not construct them from the rationals) and covers convergence, completeness, differentiation, Riemann integration, series of functions, power series, and the trigonometric and exponential functions. An unusual feature is the introduction of the Schwartz space of complex-valued functions over the reals. This long first part results in the book getting a slow start on Fourier analysis itself, but I think in most courses it could be covered fairly quickly because the use of complex-valued functions is the only aspect the student will not have seen before.
Part 2 of the book is the Fourier series part, although it also has a lot about function spaces because the approach is to show that \( L^{2}(S^{1}) \) is a Hilbert space (where \( S^{1} \) is the circle) and that the complex exponential functions \( \exp(2 \pi \imath nx) \) are an orthonormal basis. This approach requires dealing with the Lebesgue integral, and this is done in an axiomatic way where we hypothesize all the needed properties of the Lebesgue integral without actually constructing it. There is a good discussion of sets of measure zero, but no attempt to generalize to measurable sets. There are a few applications given; these include the Weierstrass approximation theorem, a simple version of the Wiener–Khinchin theorem, and an example of a continuous nowhere-differentiable function.
Part 3 is mostly about partial differential equations, and deriving those from physical problems and studying their solutions is the emphasis, rather than Fourier series. The discussion also ventures into other sorts of orthogonal systems, such as Legendre polynomials and Hermite polynomials. There’s a very focussed discussion of Hilbert spaces and operators on them. The book does not assume prior knowledge of linear algebra, so the discussion of spaces here is ad hoc toward PDEs rather than general. The applications include the vibrating string and the Schrödinger wave equation.
Part 4 is a very concise treatment of the Fourier integral. It develops the theory for the Schwartz space on the reals, and then extends this to \( L^{2} \). It does not cover the \( L^{1} \) theory. There are several applications, including the explicit solution of linear differential equations (similar to the Laplace transform method), a treatment of the heat and wave equations, a lengthy discussion of quantum observables, the Nyquist–Shannon sampling theorem, and a continuous version of the Wiener–Khinchin theorem. This section also includes a useful “what’s next?” chapter showing various directions to continue the study of Fourier analysis. The book includes a wide-ranging bibliography of works on applications and more advanced texts in analysis.
The production quality is good. I spotted only a modest number of typos, and most would not be confusing. Many of the graphs plot several functions (such as the target function and several approximations to it), and are sometimes difficult to read because the functions are not labeled in the graph.
The concept explained in the Introduction and the execution are both good, and this seems like a reasonable approach for the second course in analysis if your curriculum gives you the freedom to choose this.  The book emphasizes breadth over depth and touches lightly on a large number of topics. The author and his colleagues have used earlier drafts of this book at San Jose State University.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web site is His mathematical interests are number theory and classical analysis.