Fourier Series

This page is a sub-page of our page on Calculus of One Real Variable.

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Related KMR-pages:

Fourier Transforms

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Other relevant sources of information:

What is a Fourier series (Explained by drawing circles)
Fourier Series
Fourier Series Grapher
But what is a Fourier series? From heat flow to circle drawings

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Books:

Fourier Series and Boundary Value Problems, Brown and Churchill, McGraw-Hill, 1963.

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The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.

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Representation and Reconstruction of a Presentant with respect to a Background

Representation: \, [ \, p_{resentant} \, ]_{B_{ackground}} \, \mapsto \, \left< \, r_{epresentant} \, \right>_{B_{ackground}}

Reconstruction: \, \left( \, \left< \, r_{epresentant} \, \right>_{B_{ackground}} \, \right)_{B_{ackground}} \mapsto \,\, p_{resentant}

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\, [ \, R_{eal}V_{ector}S_{pace} \, ]_{B_{asis}} \, = \, \left< \, \mathbb{R}^n \, \right>_{B_{asis}} \, = \, \left< \, (x_1, \cdots , x_n) \, \right>_{(e_1, \cdots , \, e_n)} \,

Let \, V \, be a real vector space of dimension \, n . The fact that the vector \, x \in V \,
has the coordinates \, (x_1, \cdots , x_n) \in {\mathbb{R}}^n \, in the basis \, (e_1, \cdots , e_n) \, for \, V \,
means that each \, x \in V \, is uniquely expressible as the linear combination

\, x \, \equiv \, x_1 e_1 + \cdots + x_n e_n .

The terms of this sum are the components of the vector \, x \,
aligned with the basis \, (e_1, \cdots , e_n) \, for \, V , and
the coefficients of this linear combination are the coordinates of the vector \, x \,
with respect to this basis.

Inner product

Let \, x, y \in \mathbb{R}^n . The vector space \, \mathbb{R}^n \, can be equipped with an inner product:

\, x \cdot y \equiv (x_1 e_1 + \cdots + x_n e_n) \cdot (y_1 e_1 + \cdots + y_n e_n) ,

and, since each inner product is multilinear, and since the standard basis vectors \, (e_1, \cdots , e_n) \, are orthogonal to each other and have unit length, we have:

\, x \cdot y = x_1 y_1 + \cdots + x_n y_n \, .

Terminology: A basis with mutually orthogonal vectors is called an orthogonal basis.
If, moreover, each basis vector also has unit length, the basis is called orthonormal.

Important: In case of an orthonormal basis, each coordinate is directly computable through the inner product \, x_n = x \cdot e_n .

Hence, for an orthonormal basis, we have

\, x \, \equiv \, (x \cdot e_1) e_1 + \cdots + (x \cdot e_n) e_n .

In contrast, in the case of a non-orthogonal basis, we need to solve
a system of \, n \, linear equations in \, n \, unknowns
in order to compute the coordinate \, x_n .

Periodic functions

Let \, F \, be the set of 2π-periodic functions from \, \mathbb {R} \, to \, \mathbb {R} , let \, f, g \in F \, and let \, \lambda \in \mathbb {R} . Define the following operations (addition and scaling) on the members of \, F \, by

\, (f + g)(x) \, \stackrel {\mathrm{def}}{=} \, f(x) + g(x) \, ,

\, (\lambda f)(x) \, \stackrel {\mathrm{def}}{=} \, \lambda f(x) .

Exercise: Verify that with these definitions \, F \, becomes a vector space.

Note: This is true also for non-periodic functions,
since the periodicity of the functions is NOT used in the verification.

The \, T_{rig} \, basis for the vector space \, F

Let us define

\, T_{rig} \stackrel {\mathrm{def}}{=} (({\cos}_1, \cdots , \, {\cos}_m , \cdots) , ({\sin}_1, \cdots , \, {\sin}_n, \cdots))

where

\, {\cos}_m(x) \stackrel {\mathrm{def}}{=} \cos(m x) \, and \, {\sin}_n(x) \stackrel {\mathrm{def}}{=} \sin(n x) .

It can be shown that \, T_{rig} \, is a basis for \, F .
Moreover, with the proper inner product, \, T_{rig} \, is an orthonormal basis for \, F .

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\, [ \, f \, ]_{T_{rig}} \, = \, \left< \, ((c_1, \cdots , c_m , \cdots) , (s_1, \cdots , s_n , \cdots)) \, \right>_{T_{rig}} \,

The vector space \, F \, has infinite dimension. In fact, all periodic function spaces have the same “size” (= number of linearly independent elements) as the size (= the cardinality) of the natural numbers \, \mathbb{N} \, . This is expressed by saying that a periodic function space is countably infinite.

In contrast, a non-periodic real-to-real function space is non-countably infinite
which means that its dimensions cannot be counted by using natural numbers.

The fact that the function \, f \in F \,
has the coordinates \, ((c_1, \cdots , c_m, \cdots) , (s_1, \cdots , s_n, \cdots)) \, in the basis \, T_{rig} \,
means that

\, f(x) \, = \, c_1 {\cos}(x) + \cdots + c_m {\cos}(mx) + \cdots + \,
\qquad \quad + \, s_1 {\sin}(x) + \cdots + s_n {\sin}(nx) + \cdots \, .

The terms of this series (= infinite sum) are the components of the vector \, f \,
aligned with the basis \, T_{rig} , and
the coefficients of this linear combination are the coordinates of the vector \, f \,
with respect to this basis.

Inner product

The vector space \, F \, can be turned into an inner product space
by defining, for each pair of vectors \, f, g, \in F , an inner product

\, f \cdot g \, \stackrel {\mathrm{def}}{=} \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \, g(x) \, dx .

Exercise: Verify that this definition satisfies the axioms for an inner product on \, F .

With this inner product, we have:

\, {\cos}_m \cdot {\cos}_n \, \equiv \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} {\cos}(m x) \, {\cos}(n x) \, dx \, = \, {\delta}_{m n} ,
\, {\cos}_m \cdot {\sin}_n \, \equiv \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} {\cos}(m x) \, {\sin}(n x) \, dx \, = \, {\delta}_{m n} ,
\, {\sin}_m \cdot {\sin}_n \, \equiv \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} {\sin}(m x) \, {\sin}(n x) \, dx \, = \, {\delta}_{m n} ,

where \, {\delta}_{m n} = 1 \, if \, m = n \, and \, {\delta}_{m n} = 0 \, if \, m \neq n .

Hence \, T_{rig} \, is an orthonormal basis for the vector space \, F ,
and thus the coordinates of a vector \, f \in F \, are directly computable
through the inner products

\, c_n \, = \, f \cdot {\cos}_n \, = \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \, {\cos}(n x) \, dx ,
\, s_n \, = \, f \cdot {\sin}_n \, = \, \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x) \, {\sin}(n x) \, dx .

Hence, since \, T_{rig} \, is an orthonormal basis, we can conclude that

\, f \, \equiv \, (f \cdot {\cos}_1) \, {\cos}_1 + \cdots + (f \cdot {\cos}_m) \, {\cos}_m + \cdots + \,
\qquad + \, (f \cdot {\sin}_1) \, {\sin}_1 + \cdots + (f \cdot {\sin}_n) \, {\sin}_n + \cdots ,

or, equivalently:

\, f(x) \, \equiv \, (f \cdot {\cos}_1) \cos(x) + \cdots + (f \cdot {\cos}_m) \cos(m x) + \cdots + \,
\qquad \quad + \, (f \cdot {\sin}_1) \sin(x) + \cdots + (f \cdot {\sin}_n) \sin(n x) + \cdots .

We can combine the finite parts of the cos-series and the sin-series into
the \, N :th partial sum of the Fourier series for the function \, f \, :

\, f_N(x) \, \stackrel {\mathrm{def}}{=} \, \dfrac {c_0}{2} + \displaystyle\sum_{n=1}^N \, [ \, c_n \cos (n x) + s_n \sin (n x) \, ] .

The question is now whether (and in what sense) \, \lim\limits_{N \to \infty} f_N = f .

Convergence of \, f_N \, towards \, f \,

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Discuss the different types of regularities of the functions \, f \in F , regularities that lead to different forms of convergence, such as: point-wise convergence, \, L^2 -convergence, and convergence almost everywhere.
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Fourier series

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But what is a Fourier Series? – From heat flow to circle drawings
(3Blue1Brown on YouTube):

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Fourier series of a sawtooth wave:

Fourier Series Animation of a Saw Tooth Wave:


Brek Martin on YouTube (20 June 2018)

Fourier series of a square wave:

Odd wave function: \, f_N(x) \, \stackrel {\mathrm{def}}{=} \, \dfrac{2}{\pi} \displaystyle\sum_{n=1}^N \, (-1)^{n-1} \dfrac {2}{2n-1} \sin ((2n-1)x) \, .

Input to the Fourier Series Grapher (copy-paste): (2/pi)*(2/(2n-1))*sin((2n-1)*x)

Even wave function: \, g_N(x) \, \stackrel {\mathrm{def}}{=} \, \dfrac{2}{\pi} \displaystyle\sum_{n=0}^N \, (-1)^n \dfrac {2}{2n+1} \cos ((2n+1)x) \, .

Input to the Fourier Series Grapher (copy-paste): ((-1)^n)*(2/pi)*(2/(2n+1))*cos((2n+1)*x)
Remember to change the starting value of \, n \, in the sum from \, 1 \, to \, 0 \, .

Cosine series of even overtones: \, \displaystyle\sum_{n=1}^\infty cos(2nx)/(n^2) \,
Fourier Series Grapher (copy-paste): cos(2n*x)/(n^2)

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\, \begin{matrix} V_{ector}S_{pace}: & & \mathbb{C}^n & \mathbb{F}_{[0, 2\pi]} & \mathbb{F}_{\mathbb{R}} \\ A_{ddition}: & & \begin{matrix} x_1 \\ \vdots \\ x_n \end{matrix} + \begin{matrix} y_1 \\ \vdots \\ y_n \end{matrix} = \begin{matrix} x_1 + y_1 \\ \vdots \\ x_n + y_n \end{matrix} & (f + g)(x) = f(x) + g(x) & (f + g)(x) = f(x) + g(x) \, \end{matrix} \,

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Brek Martin on YouTube (22 Oct 2015)

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Analysis and Synthesis of Waves

A standing wave:

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