A **Fourier series** is the expansion of a [function](/page/Function) in terms of a complete orthonormal system, expressing the function as an infinite sum of basis elements weighted by its Fourier coefficients. The theory splits into two layers: the abstract [Hilbert space](/page/Hilbert%20Space) framework (which works for any ONS) and the concrete analysis of specific systems (where the choice of basis interacts with calculus, convergence, and applications). [motivation] ## Motivation ### The Abstract Layer The [Orthogonal System](/page/Orthogonal%20System) page develops Fourier series in full generality. Given any orthonormal system $\{e_k\}$ in a Hilbert space $H$, each element $x \in H$ has Fourier coefficients $c_k = (x, e_k)$ and a formal Fourier series $\sum c_k e_k$. [Bessel's inequality](/theorems/540) bounds $\sum c_k^2 \leq \|x\|^2$, the [completeness characterisation](/theorems/541) identifies when Parseval's equality holds, and the [$\ell^2$ isomorphism](/theorems/544) shows that every [separable](/page/Separable) Hilbert space is abstractly the same [sequence](/page/Sequence) space. This machinery is indifferent to the choice of ONS — it works for Legendre polynomials, Hermite functions, wavelets, or any other orthonormal system. ### The Concrete Layer But the *choice* of ONS matters enormously for applications. Different bases diagonalise different operators, interact differently with smoothness, and have different convergence behaviour. The most important concrete systems are: **Trigonometric system** $\{e^{inx}\}_{n \in \mathbb{Z}}$ on the circle $\mathbb{T}$. This is the classical Fourier series, and the subject of the [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) page. It diagonalises the [derivative](/page/Derivative) operator ($\frac{d}{dx} e^{inx} = in \, e^{inx}$), making it the natural basis for PDEs on periodic domains. The convergence theory is deep and subtle: $L^2$ convergence is guaranteed by [completeness](/theorems/585), but pointwise convergence requires additional tools — the [Dirichlet](/theorems/581) and [Fejér](/theorems/584) kernels, the [Riemann-Lebesgue lemma](/theorems/245), and [Dini's criterion](/theorems/583). **Passing to the line.** When the period $L \to \infty$, the discrete frequencies $n/L$ become continuous, the Fourier coefficients become the Fourier transform, and the [series](/page/Series) becomes an integral. This passage — from Fourier series to [Fourier transform](/page/Fourier%20Transform) — is the subject of the [Fourier Transform](/page/Fourier%20Transform) page. ### What This Page Is This page is a hub: it describes the landscape of Fourier series and directs the reader to the appropriate child page. The abstract theory lives on the [Orthogonal System](/page/Orthogonal%20System) page; the concrete trigonometric theory lives on the [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) page; and the non-periodic analogue is on the [Fourier Transform](/page/Fourier%20Transform) page. [/motivation] ## The General Setup Let $H$ be a separable Hilbert space with inner product $(\cdot\,,\cdot)_H$, and let $\{e_k\}_{k \in I}$ be an orthonormal system indexed by a [countable set](/page/Countable%20Set) $I$. For each $x \in H$, the **Fourier coefficients** are $c_k(x) := (x, e_k)_H$, and the **Fourier series** is the formal sum $\sum_{k \in I} c_k(x) \, e_k$. The central questions are: **Convergence.** Does the series converge to $x$ in the norm of $H$? By the [completeness characterisation](/theorems/541), this happens for all $x$ if and only if the ONS is complete (its closed span is all of $H$). Completeness must be verified separately for each system. **Parseval's identity.** When the system is complete, the norm is computed from the coefficients: $\|x\|_H^2 = \sum |c_k(x)|^2$. This is the Hilbert space version of the Pythagorean theorem, extended to infinitely many terms. **Pointwise convergence.** In function spaces like $L^2(\mathbb{T})$ or $L^2(\mathbb{R})$, one can ask whether the series converges not just in norm but pointwise (at specific values of the argument). This question is much harder than norm convergence and depends on the specific ONS and the regularity of the function. For the trigonometric system, the pointwise theory is developed on the [trigonometric page](/page/Fourier%20Series%20(Trigonometric)). ## Child Pages ### [Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric)) The classical theory of Fourier series on the circle. Topics: Dirichlet and Fejér kernels, pointwise convergence criteria (Dini, Dirichlet-Jordan), Cesàro summability, completeness of the trigonometric system, Parseval computations ($\sum 1/n^2 = \pi^2/6$), smoothness-decay duality, and applications to the heat equation. **Key theorems:** [Dirichlet Kernel Formula](/theorems/581), [Riemann-Lebesgue Lemma](/theorems/245), [Dini's Criterion](/theorems/583), [Fejér's Theorem](/theorems/584), [Completeness](/theorems/585), [Decay under Differentiability](/theorems/586), [Smoothness from Decay](/theorems/587). ### [Fourier Transform](/page/Fourier%20Transform) The non-periodic analogue: Fourier analysis on $\mathbb{R}^n$. Discrete frequencies are replaced by a [continuous](/page/Continuity) frequency variable $\xi$, sums become [integrals](/page/Integral), and the Plancherel theorem replaces Parseval. The Fourier transform diagonalises constant-coefficient differential operators and is the foundational tool for PDE theory on $\mathbb{R}^n$. ## Related Pages - [Orthogonal System](/page/Orthogonal%20System) — the abstract Hilbert space theory (Bessel, Parseval, completeness, $\ell^2$ isomorphism) - [Sobolev Space](/page/Sobolev%20Space) — Fourier characterisation of Sobolev regularity ($f \in H^s \iff \sum |n|^{2s}|\hat{f}(n)|^2 < \infty$) - [Heat Equation](/page/Heat%20Equation) — Fourier series solve the heat IBVP on periodic domains; each mode decays as $e^{-n^2 t}$ - [Laplace's Equation](/page/Laplace's%20Equation) — the Poisson integral on the disk is the Fourier series evaluated at radius $\rho < 1$ ## References - Katznelson, Y. (2004). *An Introduction to Harmonic Analysis* (3rd ed.). Cambridge University Press. - Stein, E. M. and Shakarchi, R. (2003). *Fourier Analysis: An Introduction*. Princeton University Press. - Grafakos, L. (2014). *Classical Fourier Analysis* (3rd ed.). Springer.