Fourier Analysis: Basic
First- and second-year university level
Overview
The Basic level introduces a rigorous definition of Fourier series and the systematic computation of Fourier coefficients. It develops the expansions of even and odd functions, Parseval's identity, and the convergence theorems — the foundations of Fourier analysis.
Prerequisites
- The Introduction level material
- Basic calculus (substitution and integration by parts)
- Familiarity with elementary series convergence
Chapters
Learning Goals
- Compute Fourier series for functions of any period
- Exploit symmetry to simplify the computation
- Understand the relationship between complex and real Fourier series
- Use Parseval's identity to evaluate infinite series
- State the types of convergence and the conditions for each
Frequently Asked Questions
What is covered in the Basic level of Fourier Analysis?
Orthogonality of trigonometric functions, computation of Fourier coefficients, basic Fourier expansions (square wave, sawtooth, etc.), and even/odd function expansions. The complex exponential form, Parseval's identity and Bessel's inequality, convergence theorems, the Dirichlet and Fejér kernels, the Gibbs phenomenon, and an introduction to the Fourier transform are also included — 11 chapters in total.
How are Fourier coefficients computed?
For a function $f(x)$ of period $2\pi$, the Fourier coefficients are $a_n = \dfrac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx$ and $b_n = \dfrac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx$. The orthogonality relation $\int_{-\pi}^{\pi}\cos(mx)\cos(nx)\,dx = \pi\delta_{mn}$ is the key tool.
For which functions does the Fourier series converge?
Fourier series are defined for piecewise-smooth periodic functions (differentiable on each piece, with finitely many discontinuities). The series converges to the function value at continuous points and to the average of the left and right limits at jumps (Dirichlet's theorem). $L^2$ convergence holds on a broader class of functions.