- Complex valued-functions and the complex exponential. Fourier coefficients of piecewise continuous periodic functions. Basic operations and their effects on Fourier coefficients: translation, modulation, convolutions, derivatives.
- Uniform convergence: Cesaro means, the Dirichlet and Fejer kernels, Fejer’s theorem. The Weierstrass approximation theorem for trigonometric polynomials and for polynomials. Uniqueness of Fourier coefficients. The Riemann-Lebesgue lemma. Hausdorff’s moment problem. Convergence of partial sums and Fourier series for $C^2$-functions.
- Pointwise convergence: Dini’s criterion. Convergence at jump discontinuities and Gibbs phenomenon.
- $L^2$-theory: orthonormal sequences and bases. Best approximations, Bessel’s inequality, Parseval’s identity and convergence in $L^2$.
- Applications to partial differential equations: the heat and wave equations on an interval with constant boundary conditions, the Dirichlet problem for the Laplace equation on the disk, the Poisson kernel.
- Korner, Fourier analysis
- Stein and Shakarchi, Fourier analysis