Chapter 1. The Genesis of Fourier Analysis
Chapter 2. Basic Properties of Fourier Series
Chapter 3. Convergence of Fourier Series
Chapter 4. Some Applications of Fourier Series
Chapter 5. The Fourier Transform on ℝ
Chapter 6. The Fourier Transform on ℝd
Chapter 7. Finite Fourier Analysis
Chapter 8. Dirichlet’s Theorem

This first volume, a three-part introduction to the subject, is intended for students with a beginning knowledge of mathematical analysis who are motivated to discover the ideas that shape Fourier analysis. It begins with the simple conviction that Fourier arrived at in the early nineteenth century when studying problems in the physical sciences—that an arbitrary function can be written as an infinite sum of the most basic trigonometric functions.

The first part implements this idea in terms of notions of convergence and summability of Fourier series, while highlighting applications such as the isoperimetric inequality and equidistribution. The second part deals with the Fourier transform and its applications to classical partial differential equations and the Radon transform; a clear introduction to the subject serves to avoid technical difficulties. The book closes with Fourier theory for finite abelian groups, which is applied to prime numbers in arithmetic progression. --- summary provided by publisher