Suitable for advanced undergraduates and graduate students, this text develops comparison theorems to establish the fundamentals of Fourier analysis and to illustrate their applications to partial differential equations.

The three-part treatment begins by establishing the quotient structure theorem or fundamental principle of Fourier analysis. Topics include the geometric structure of ideals and modules, quantitative estimates, and examples in which the theory can be applied. The second part focuses on applications to partial differential equations and covers the solution of homogeneous and inhomogeneous systems, existence and uniqueness questions related to Cauchy's problem, and boundary value problems for solutions in a cube. The final section explores functions and their role in Fourier representation. Each chapter begins with a detailed summary, and most conclude with general remarks, bibliographical remarks, and problems for further study.