Math 644: Harmonic Analysis
Harmonic analysis on the torus and in Euclidean space; pointwise and norm convergence of Fourier series and functional-analytic aspects of Fourier transforms emphasized.
Desired Learning Outcomes
Minimal learning outcomes
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, i.e., convolution.
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. How to differentiate distributions. The Fourier transform of distributions.
Application of the Fourier transform to differential equations. In particular we will discuss the heat equation and the wave equation.
Hermite functions and polynomials.
Other integral transforms. In particular, we will discuss the continuous wavelet transform, derive a Plancherel formula and an inversion formula.
Other relevant topics may also be covered.
Possible textbooks for this course include (but are not limited to):