Students should know all relevant definitions, correct statements of the major theorems (including their hypotheses and limitations), and examples and non-examples of the various concepts. The students should be able to demonstrate their mastery by solving non-trivial problems related to these concepts, and by proving simple (but non-trivial) theorems about the concepts below, related to, but not identical to, statements proven by the text or instructor. For more detailed information visit the Math 644 Wiki page.

Periodic functions and Fourier series; Convergence of Fourier series; Spaces of functions on R^n

The space of compactly supported functions, functions of compact support and the algebraic structure of those spaces.

The Fourier transform of rapidly decreasing functions and L^2 functions, inversion formula and Plancherel theorems.

Introduction to distribution theory and the continuous linear functionals on function spaces. The Fourier transform of distributions.

Application of the Fourier transform to differential equations.

Hermite functions and polynomials.

Other integral transforms.