Beijing Institute of Mathematical Sciences and Applications

We begin with Fourier series, examining their properties and essential results, including best square approximation, the Dirichlet kernel, convolutions, and convergence. These concepts are applied to solve key PDEs, such as the Laplace, heat, and wave equations. We then introduce the Fourier transform, a powerful tool for tackling PDEs in higher dimensions without the requirement of periodicity. Key foundational results are established, including the Fourier inversion formula, Plancherel's theorem, and the approximation of identity in ${\mathbb{R}}^n$. We also investigate "good kernels," such as the heat and Poisson kernels. Using the Fourier transform, we provide solutions for the Laplace equation in the upper half-space and address the Cauchy problem for the heat equation. Additionally, we explore the Heisenberg uncertainty principle. The course further delves into boundary value problems for the Laplace equation in higher-dimensional domains, deriving representation formulas and studying fundamental solutions, the mean value property, Harnack’s inequality, and maximum principles.