Fourier Series

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Source: algebrica.org - CC BY-NC 4.0 https://algebrica.org/fourier-series/ Fetched from algebrica.org post 22263; source modified 2026-03-14T21:51:52.

Definition

A Fourier series represents a periodic function as an infinite sum of sine and cosine functions. More precisely, it shows that periodic behavior can be decomposed into elementary harmonic oscillations. This result expresses a structural property of periodic functions: oscillatory components form a natural coordinate system for describing repetition.

Let $f : \mathbb{R} \to \mathbb{R}$ be a function that is periodic with period $2\pi$, meaning:

f(x + 2\pi) = f(x) \, \forall x \in \mathbb{R}

Assume that $f$ is integrable on the interval $[-\pi,\pi]$. The Fourier series of $f$ is the formal trigonometric expansion:

f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx)
  • The symbol $\sim$ emphasizes that we are not yet asserting equality (we are defining a trigonometric series associated with $f$).
  • The question of whether the series converges to $f$ will be addressed later.
  • Each term $\cos(nx)$ and $\sin(nx)$ represents an oscillation of frequency $n$.
  • The expansion therefore decomposes $f$ into its harmonic components.

Fourier Coefficients

The coefficients $a_n$ and $b_n$ are defined by the following integrals:

\begin{align} a_0 &= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\,dx \\\\ a_n &= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\cos(nx)\,dx \\\\ b_n &= \frac{1}{\pi} \int_{-\pi}^{\pi} f(x)\sin(nx)\,dx \quad n \ge 1 \end{align}

These expressions are not introduced by convention, and they are not chosen just because they work. They follow from a structural fact about sines and cosines: over a full period they are orthogonal to one another. On the interval $[-\pi,\pi]$, trigonometric waves with different frequencies remain independent under integration, which is exactly what lets us isolate one harmonic at a time and read off the corresponding coefficient.

\begin{align} \int_{-\pi}^{\pi} \cos(nx)\cos(mx)\,dx &= \begin{cases} \pi & n=m\neq 0 \\\\ 0 & n\ne m \end{cases} \\[6pt] \int_{-\pi}^{\pi} \sin(nx)\sin(mx)\,dx &= \begin{cases} \pi & n=m \\\\ 0 & n\ne m \end{cases} \\[6pt] \int_{-\pi}^{\pi} \sin(nx)\cos(mx)\,dx &= 0 \end{align}

These relations imply that the trigonometric system behaves like an orthogonal basis under the inner product:

\langle f, g \rangle = \int_{-\pi}^{\pi} f(x)g(x)\,dx

Each coefficient measures how much of a specific harmonic direction is present in the function. In this sense, the Fourier expansion is a projection process in an infinite-dimensional space.

Example 1

This example illustrates how even a simple linear function acquires a rich harmonic structure when periodically extended. Consider the function $f(x) = x$, defined on $(-\pi,\pi)$ and extended periodically with period $2\pi$. This function is odd. Therefore:

a_0 = 0 \quad a_n = 0

We compute the sine coefficients:

b_n = \frac{1}{\pi} \int_{-\pi}^{\pi} x\sin(nx)\,dx

Using integration by parts, we obtain:

b_n = \frac{2(-1)^{n+1}}{n}

Hence the Fourier series is:

x \sim 2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin(nx)
The coefficients decay like $\frac{1}{n}$. The slower decay reflects the fact that $f$ is continuous but not differentiable at the endpoints of the period. The periodic extension introduces jump discontinuities at multiples of $\pi$, which influences convergence behavior.

Convergence of Fourier series

The definition of the Fourier series does not automatically guarantee convergence to the original function. A classical result states that if $f$ satisfies the following Dirichlet conditions:

  • $f$ is piecewise continuous
  • $f$ has finitely many local extrema in $[-\pi,\pi]$
  • $f$ has finitely many jump discontinuities

then the Fourier series converges at every point $x.$ More precisely consider the $N$-th partial sum:

S_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N} a_n\cos(nx)+b_n\sin(nx)
\lim_{N\to\infty} S_N(x) = \frac{f(x^+)+f(x^-)}{2}

At points where $f$ is continuous, the series converges to $f(x)$. At jump discontinuities, it converges to the midpoint of the left and right limits. This behavior reveals a fundamental property of Fourier approximation: it respects average local behavior rather than pointwise values at discontinuities.

  • If $f$ is continuously differentiable, coefficients decay faster.
  • If $f$ has discontinuities, decay is slower.
  • The smoother the function, the more rapidly the harmonic amplitudes decrease.

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