Fourier Series

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...
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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 $f : \mathbb{R} \rightarrow \mathbb{R}$ 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:

\[f ( x ) sim \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} a_{n} cos ⁡ ( n x ) + b_{n} sin ⁡ ( n x )\]
  • 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 $cos ⁡ ( n x )$ and $sin ⁡ ( n x )$ 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:

\[a_{0} & = \frac{1}{\pi} \int_{- \pi}^{\pi} f ( x ) d x \\ a_{n} & = \frac{1}{\pi} \int_{- \pi}^{\pi} f ( x ) cos ⁡ ( n x ) d x \\ b_{n} & = \frac{1}{\pi} \int_{- \pi}^{\pi} f ( x ) sin ⁡ ( n x ) d x n \geq 1\]

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.

\[\int_{- \pi}^{\pi} cos ⁡ ( n x ) cos ⁡ ( m x ) d x & = \{ \begin{matrix}\pi & n = m \neq 0 \\ 0 & n \neq m\end{matrix} \\ \int_{- \pi}^{\pi} sin ⁡ ( n x ) sin ⁡ ( m x ) d x & = \{ \begin{matrix}\pi & n = m \\ 0 & n \neq m\end{matrix} \\ \int_{- \pi}^{\pi} sin ⁡ ( n x ) cos ⁡ ( m x ) d x & = 0\]

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

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

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:

\[a_{0} = 0 a_{n} = 0\]

We compute the sine coefficients:

\[b_{n} = \frac{1}{\pi} \int_{- \pi}^{\pi} x sin ⁡ ( n x ) d x\]

Using integration by parts, we obtain:

\[b_{n} = \frac{2 ( - 1 )^{n + 1}}{n}\]

Hence the Fourier series is:

\[x sim 2 \sum_{n = 1}^{\infty} \frac{( - 1 )^{n + 1}}{n} sin ⁡ ( n x )\]
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:

\[S_{N} ( x ) = \frac{a_{0}}{2} + \sum_{n = 1}^{N} a_{n} cos ⁡ ( n x ) + b_{n} sin ⁡ ( n x )\] \[\underset{N \rightarrow \infty}{lim} S_{N} ( x ) = \frac{f ( x^{+} ) + f ( x^{-} )}{2}\]

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.

Selected references

Next: Part 14.
Functions
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Taylor Series