A primer on Fourier Series

TL;DR

The work surveys Fourier Series foundations for $1$-periodic functions on $[0,1)$ by treating Fourier coefficients as $L^2$-projections onto the complex exponential basis $\{e^{2\pi i kx}\}$ and by formalizing best $L^2$-approximations via trigonometric polynomials. It then develops the core convergence theory, including the Riemann-Lebesgue lemma, the Dirichlet and Fejér kernels, and pointwise as well as $L^2$ and $L^p$ convergence results, together with localization, uniqueness, and the decay-regularity relationships. The text also presents a cohesive narrative around the Dirichlet kernel, Dini-type conditions for pointwise convergence, and the Hilbert-space viewpoint that yields Parseval and the Riesz-Fischer theorem, while highlighting limitations (e.g., $L^1$-case and jump discontinuities) and the role of smoothness in decay. Overall, it ties Fourier coefficients, convergence modes, and function regularity into a unified framework with extensive exercises and solutions to reinforce intuition and rigor.

Abstract

This is a textbook on Fourier Series, suitable for both undergraduate and graduate courses. The textbook is endowed with exercises, and full solutions are provided at the end of the book.

Figures (70)

• Figure 1: Periodic extensions of the functions $f(x)=x$, $f(x)=x^2$, $f(x)=x(1-x)$, and $f(x)=\sin(2\pi x)$.
• Figure 2: Plot of a square wave approximation $\sum_{j=0}^{N} \frac{4}{\pi(2j+1)}\sin(2\pi(2j+1)x)$, with $N\in\{5,20,50\}$.
• Figure 3: Plot of a sawtooth wave approximation $-\sum_{k=1}^{N}\frac{1}{\pi k}\sin(2\pi kx)$, with $N\in\{5,20,50\}$.
• Figure 4: The Dirichlet Kernel (left, with $N=5$; right, with $N=20$).
• Figure 5: The Fejér Kernel (left, with $N=5$; right, with $N=20$).

Theorems & Definitions (102)

• Theorem 2.2.1
• Lemma 2.2.2
• proof
• Lemma 2.2.3
• proof
• Corollary 2.2.4
• proof
• Corollary 2.2.5
• proof
• proof : Proof of Theorem \ref{['BEST']}