# Fourier Series

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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](../sine-function.md) and [cosine functions](../cosine-function.md). More precisely, it shows that periodic behavior can be decomposed into elementary harmonic oscillations. This result expresses a structural property of periodic [functions](../functions.md): 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](../definite-integrals.md) on the [interval](../intervals.md) \\( \[-\\pi,\\pi\] \\). 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](../sine-and-cosine.md): 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](../sine-and-cosine.md) coefficients:

\\[
b\_n = \\frac{1}{\\pi} \\int\_{-\\pi}^{\\pi} x\\sin(nx)\\,dx
\\]

Using [integration by parts](../integration-by-parts.md), 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](../dirichlet-function.md):

-   \\( 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](../limits.md). 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

-   **E. M. Stein, R. Shakarchi**. [Fourier Analysis: An Introduction](https://kryakin.site/am2/Stein-Shakarchi-1-Fourier_Analysis.pdf)
-   **L. Grafakos**. [Classical Fourier Analysis](https://www.math.stonybrook.edu/~bishop/classes/math638.F20/Grafakos_Classical_Fourier_Analysis.pdf)
-   **G. P. Tolstov**. [Fourier Series](https://archive.org/embed/tolstov-fourier-series-1962)
-   **G. B. Folland**. [Fourier Analysis and Its Applications](https://www-elec.inaoep.mx/~rogerio/Tres/FourierAnalysisUno.pdf)
-   **A. Zygmund**. [Trigonometric Series](https://archive.org/details/trigonometricseries)
-   **Y. Katznelson**. [An Introduction to Harmonic Analysis](https://archive.org/details/introductiontoha0000katz)
-   **Stanford University**. [The Fourier Transform and Its Applications](https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf)
-   **Oxford University Press**. [Fourier Series and Fourier Transforms](https://academic.oup.com/book/54863/chapter/422699978)
-   **R. Herman**. [An Introduction to Fourier and Complex Analysis](https://people.uncw.edu/hermanr/mat367/fcabook/FCA)