The first four Fourier series approximations for a square wave.

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

The Fourier series is named in honour of Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur in 1822.

The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.

From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,^[1] thin-walled shell theory,^[2] etc.

Revolutionary article[link]

to Failed to parse (Missing texvc executable; please see math/README to configure.): y=+1
yields:

This immediately gives any coefficient Failed to parse (Missing texvc executable; please see math/README to configure.): a_k

of the trigonometrical series for Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi(y)
for any function which has such an expansion. It works because if Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
has such an expansion, then (under suitable convergence assumptions) the integral

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} a_k&=\int_{-1}^1\varphi(y)\cos(2k+1)\frac{\pi y}{2}\,dy \\ &= \int_{-1}^1\left(a\cos\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}\cos(2k+1)\frac{\pi y}{2}+\cdots\right)\,dy \end{align}

can be carried out term-by-term. But all terms involving Failed to parse (Missing texvc executable; please see math/README to configure.): \cos(2j+1)\frac{\pi y}{2} \cos(2k+1)\frac{\pi y}{2}

for j ≠ k vanish when integrated from −1 to 1, leaving only the kth term.

In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.

When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

Birth of harmonic analysis[link]

Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Fourier originally defined the Fourier series for real-valued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition.

Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called harmonic analysis. A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval.

Definition[link]

In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

[edit] Fourier's formula for 2π-periodic functions using sines and cosines

For a periodic function ƒ(x) that is integrable on [−π, π], the numbers

Failed to parse (Missing texvc executable; please see math/README to configure.): a_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \cos(nx)\, dx, \quad n \ge 0

and

Failed to parse (Missing texvc executable; please see math/README to configure.): b_n = \frac{1}{\pi}\int_{-\pi}^\pi f(x) \sin(nx)\, dx, \quad n \ge 1

are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by

Failed to parse (Missing texvc executable; please see math/README to configure.): (S_N f)(x) = \frac{a_0}{2} + \sum_{n=1}^N \, [a_n \cos(nx) + b_n \sin(nx)], \quad N \ge 0.

The partial sums for ƒ are trigonometric polynomials. One expects that the functions S_N ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)]

is called the Fourier series of ƒ. These trigonometric functions can themselves be expanded, using multiple angle formulae.

The Fourier series does not always converge, and even when it does converge for a specific value x₀ of x, the sum of the series at x₀ may differ from the value ƒ(x₀) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function. If a function is square-integrable on the interval [−π, π], then the Fourier series converges to the function at almost every point. In engineering applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to this presumption. In particular, the Fourier series converges absolutely and uniformly to ƒ(x) whenever the derivative of ƒ(x) (which may not exist everywhere) is square integrable.^[4] See Convergence of Fourier series.

It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.

Example 1: a simple Fourier series[link]

Plot of a periodic identity function—a sawtooth wave.

Animated plot of the first five successive partial Fourier series.

We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x, \quad \mathrm{for } -\pi < x < \pi,

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x + 2\pi k) = f(x), \quad \mathrm{for } -\infty < x < \infty \text{ and } k \in \mathbb{Z} .

In this case, the Fourier coefficients are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} a_0 &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x\,dx = 0. \\ a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0, \quad n \ge 0. \\ b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) + \frac{2}{\pi n^2}\sin(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}

It can be proven that the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, and therefore:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} f(x) &= \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\ &=2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad x - \pi \notin 2 \pi \mathbb{Z}. \end{align}

(Eq.1)

When x = π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ƒ at x = π. This is a particular instance of the Dirichlet theorem for Fourier series.

Example 2: Fourier's motivation[link]

Heat distribution in a metal plate, using Fourier's method

One notices that the Fourier series expansion of our function in example 1 looks much less simple than the formula ƒ(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates (x, y) ∈ [0, π] × [0, π]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = π, is maintained at the temperature gradient T(x, π) = x degrees Celsius, for x in (0, π), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.

Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of  Eq.1 by sinh(ny)/sinh(nπ). While our example function f(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x, y) is nontrivial. The function T cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

Other applications[link]

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ζ(2n), for any positive integer n.

Exponential Fourier series[link]

We can use Euler's formula,

Failed to parse (Missing texvc executable; please see math/README to configure.): e^{inx} = \cos(nx)+i\sin(nx), \,

where i is the imaginary unit, to give a more concise formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \sum_{n=-\infty}^ \infty c_n e^{inx}.

Assuming f(x) is a periodic function with T = 2π, the Fourier coefficients are then given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, dx.

The Fourier coefficients a_n, b_n, c_n are related via

Failed to parse (Missing texvc executable; please see math/README to configure.): a_n = { c_n + c_{-n} } \quad \text{ for }n=0,1,2,\dots\,

Failed to parse (Missing texvc executable; please see math/README to configure.): b_n = i( c_{n} - c_{-n} ) \quad \text{ for }n=1,2,\dots\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): c_n = \begin{cases} \frac{1}{2}(a_n - i b_n) & n > 0 \\ \quad \frac{1}{2}a_0 & n = 0 \\ \frac{1}{2}(a_{-n} + i b_{-n}) & n < 0. \end{cases}

The notation c_n is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of ƒ (in this case), such as Failed to parse (Missing texvc executable; please see math/README to configure.): \scriptstyle\hat{f}   or F, and functional notation often replaces subscripting:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} f(x) &= \sum_{n=-\infty}^\infty \hat{f}(n)\cdot e^{inx} \\ &= \sum_{n=-\infty}^\infty F[n]\cdot e^{jnx} \quad \text{(electrical engineering).} \end{align}

In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

[edit] Fourier series on a general interval [a, a + τ]

The following formula, with appropriate complex-valued coefficients G[n], is a periodic function with period τ on all of R:

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}.

If a function is square-integrable in the interval [a, a + τ], it can be represented in that interval by the formula above.  I.e., when the coefficients are derived from a function, h(x), as follows:

Failed to parse (Missing texvc executable; please see math/README to configure.): G[n] = \frac{1}{\tau}\int_a^{a+\tau} h(x)\cdot e^{-i 2\pi \frac{n}{\tau} x}\, dx,

then g(x) will equal h(x) in the interval [a,a+τ ]. It follows that if h(x) is τ-periodic, then:

• g(x) and h(x) are equal everywhere, except possibly at discontinuities, and
• a is an arbitrary choice. Two popular choices are a = 0, and a = −τ/2.

Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:

Failed to parse (Missing texvc executable; please see math/README to configure.): G(f) \ \stackrel{\mathrm{def}}{=} \ \sum_{n=-\infty}^\infty G[n]\cdot \delta \left(f-\frac{n}{\tau}\right),

where variable ƒ represents a continuous frequency domain. When variable x has units of seconds, ƒ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/τ, which is called the fundamental frequency.  g(x) can be recovered from this representation by an inverse Fourier transform:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \mathcal{F}^{-1}\{G(f)\} &= \int_{-\infty}^\infty \left( \sum_{n=-\infty}^\infty G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)\right) e^{i 2 \pi f x}\,df, \\ &= \sum_{n=-\infty}^\infty G[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{\tau}\right) e^{i 2 \pi f x}\,df, \\ &= \sum_{n=-\infty}^\infty G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \ \ \stackrel{\mathrm{def}}{=} \ g(x). \end{align}

The function G(ƒ) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.^{[nb 2]}

Fourier series on a square[link]

We can also define the Fourier series for functions of two variables x and y in the square [−π, π]×[−π, π]:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y) = \sum_{j,k \in \mathbb{Z}\text{ (integers)}} c_{j,k}e^{ijx}e^{iky},

Failed to parse (Missing texvc executable; please see math/README to configure.): c_{j,k} = {1 \over 4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x,y) e^{-ijx}e^{-iky}\, dx \, dy.

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, which is a Fourier transform using the cosine basis functions.

Hilbert space interpretation[link]

In the language of Hilbert spaces, the set of functions Failed to parse (Missing texvc executable; please see math/README to configure.): \{ e_n = e^{i n x},n\in\mathbb{Z}\}

is an orthonormal basis for the space Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi])
of square-integrable functions of Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi]

. This space is actually a Hilbert space with an inner product given for any two elements f and g by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle f,\, g \rangle \;\stackrel{\mathrm{def}}{=} \; \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.

The basic Fourier series result for Hilbert spaces can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): f=\sum_{n=-\infty}^\infty \langle f,e_n \rangle \, e_n.

This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1, \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1

(where Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_{mn}

is the Kronecker delta), and

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = 0 \, ;\,

furthermore, the sines and cosines are orthogonal to the constant function 1. An orthonormal basis for L²([−π, π]) consisting of real functions is formed by the functions 1, and √2 cos(n  x),  √2 sin(n x) for n = 1, 2,...  The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.

Properties[link]

We say that ƒ belongs to  Failed to parse (Missing texvc executable; please see math/README to configure.): C^k(\mathbb{T})   if ƒ is a 2π-periodic function on R which is k times differentiable, and its kth derivative is continuous.

• If ƒ is a 2π-periodic odd function, then Failed to parse (Missing texvc executable; please see math/README to configure.): a_n=0

  for all n.

• If ƒ is a 2π-periodic even function, then Failed to parse (Missing texvc executable; please see math/README to configure.): b_n=0

  for all n.

• If ƒ is integrable, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{|n|\rightarrow \infty}\hat{f}(n)=0

, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}a_n=0

and Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}b_n=0.
 This result is known as the Riemann–Lebesgue lemma.

• A doubly infinite sequence Failed to parse (Missing texvc executable; please see math/README to configure.): \{a_n\}

in Failed to parse (Missing texvc executable; please see math/README to configure.): c_0(\mathbb{Z})
is the sequence of Fourier coefficients of a function in Failed to parse (Missing texvc executable; please see math/README to configure.): L^1[0,2\pi]
if and only if it is a convolution of two sequences in Failed to parse (Missing texvc executable; please see math/README to configure.): \ell^2(\mathbb{Z})

. See [1]

• If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^1(\mathbb{T})

, then the Fourier coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f'}(n)

of the derivative Failed to parse (Missing texvc executable; please see math/README to configure.): f'
can be expressed in terms of the Fourier coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f}(n)
of the function Failed to parse (Missing texvc executable; please see math/README to configure.): f

, via the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f'}(n) = in \hat{f}(n) .

• If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^k(\mathbb{T})

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f^{(k)}}(n) = (in)^k \hat{f}(n) . In particular, since Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f^{(k)}}(n)

tends to zero, we have that Failed to parse (Missing texvc executable; please see math/README to configure.): |n|^k\hat{f}(n)
tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.

• Parseval's theorem. If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in L^2([-\pi,\pi])

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=-\infty}^\infty |\hat{f}(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2 \, dx .

• Plancherel's theorem. If Failed to parse (Missing texvc executable; please see math/README to configure.): c_0,\, c_{\pm 1},\, c_{\pm 2},\ldots

are coefficients and Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=-\infty}^\infty |c_n|^2 < \infty
then there is a unique function Failed to parse (Missing texvc executable; please see math/README to configure.): f\in L^2([-\pi,\pi])
such that Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f}(n) = c_n
for every n.

• The first convolution theorem states that if ƒ and g are in L¹([−π, π]), then Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f*g}(n) = 2\pi\hat{f}(n)\hat{g}(n)

, where ƒ ∗ g denotes the 2π-periodic convolution of ƒ and g. (The factor Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

is not necessary for 1-periodic functions.) 

• The second convolution theorem states that Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f\cdot g} = \hat{f}*\hat{g}

.

• The Poisson summation formula states that the periodic summation of a function,  Failed to parse (Missing texvc executable; please see math/README to configure.): f,

  has a Fourier series representation whose coefficients are proportional to discrete samples of the continuous Fourier transform of Failed to parse (Missing texvc executable; please see math/README to configure.): f :

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=-\infty}^\infty f(t + nT) = \frac{1}{T} \sum_{k=-\infty}^\infty \hat{f}\left(\frac{k}{T}\right)\ \exp\left(i 2\pi \frac{k}{T} t\right)

.

 

Similarly, the periodic summation of Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f}

has a Fourier series representation whose coefficients are proportional to discrete samples of Failed to parse (Missing texvc executable; please see math/README to configure.): f

, a fact which provides a pictorial understanding of aliasing and the famous sampling theorem.

• Also see Fourier_analysis#Variants_of_Fourier_analysis.

Compact groups[link]

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L²(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [−π, π] case.

An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.

Riemannian manifolds[link]

The atomic orbitals of chemistry are spherical harmonics and can be used to produce Fourier series on the sphere.

If the domain is not a group, then there is no intrinsically defined convolution. However, if X is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold X. Then, by analogy, one can consider heat equations on X. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type L²(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the [−π, π] case. A typical example is to take X to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.

Locally compact Abelian groups[link]

The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightfoward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to L¹(G) or L²(G), where G is an LCA group. If G is compact, one also obtains a Fourier series, which converges similarly to the [−π, π] case, but if G is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R} .

Approximation and convergence of Fourier series[link]

An important question for the theory as well as applications is that of convergence. In particular, it is often necessary in applications to replace the infinite series Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{-\infty}^\infty   by a finite one,

Failed to parse (Missing texvc executable; please see math/README to configure.): (S_N f)(x) = \sum_{n=-N}^N \hat{f}(n) e^{inx}.

This is called a partial sum. We would like to know, in which sense does (S_N ƒ)(x) converge to ƒ(x) as N tends to infinity.

Least squares property[link]

We say that p is a trigonometric polynomial of degree N when it is of the form

Failed to parse (Missing texvc executable; please see math/README to configure.): p(x)=\sum_{n=-N}^N p_n e^{inx}.

Note that S_N ƒ is a trigonometric polynomial of degree N. Parseval's theorem implies that

Theorem. The trigonometric polynomial S_N ƒ is the unique best trigonometric polynomial of degree N approximating ƒ(x), in the sense that, for any trigonometric polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): p\neq S_N f

of degree N, we have  Failed to parse (Missing texvc executable; please see math/README to configure.): \|S_N f - f\|_2 < \|p - f\|_2.

Here, the Hilbert space norm is

Failed to parse (Missing texvc executable; please see math/README to configure.): \| g \|_2 = \sqrt{{1 \over 2\pi} \int_{-\pi}^{\pi} |g(x)|^2 \, dx}.

Convergence[link]

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.

Theorem. If ƒ belongs to L²([−π, π]), then the Fourier series converges to ƒ in L²([−π, π]), that is,  Failed to parse (Missing texvc executable; please see math/README to configure.): \|S_N f - f\|_2

converges to 0 as N goes to infinity.

We have already mentioned that if ƒ is continuously differentiable, then  Failed to parse (Missing texvc executable; please see math/README to configure.): i n \hat{f}(n)   is the nth Fourier coefficient of the derivative ƒ′. It follows, essentially from the Cauchy–Schwarz inequality, that the Fourier series of ƒ is absolutely summable. The sum of this series is a continuous function, equal to ƒ, since the Fourier series converges in the mean to ƒ:

Theorem. If  Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^1(\mathbb{T}) , then the Fourier series converges to ƒ uniformly (and hence also pointwise.)

This result can be proven easily if ƒ is further assumed to be C², since in that case Failed to parse (Missing texvc executable; please see math/README to configure.): n^2\hat{f}(n)

tends to zero as Failed to parse (Missing texvc executable; please see math/README to configure.): n\to\infty

. More generally, the Fourier series is absolutely summable, thus converges uniformly to ƒ, provided that ƒ satisfies a Hölder condition of order α > ½. In the absolutely summable case, the inequality  Failed to parse (Missing texvc executable; please see math/README to configure.): \sup_x |f(x) - (S_N f)(x)| \le \sum_{|n| > N} |\hat{f}(n)|   proves uniform convergence.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at x if ƒ is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an L² function actually converges almost everywhere.

These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".^[5][6][7][8]

Divergence[link]

Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous T-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov published an article entitled "Une série de Fourier-Lebesgue divergente presque partout" in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere (Katznelson 1976).

See also[link]

• Gibbs phenomenon
• Laurent series — the substitution q = e^ix transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant.
• Sturm–Liouville theory
• ATS theorem
• Discrete Fourier transform
• Fast Fourier transform
• Spectral theory
• Fejér's theorem
• Dirichlet kernel

Notes[link]

1. ^ These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.
2. ^ Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as distributions. In this sense Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{F}\{e^{i2\pi \frac{n}{\tau} x}\} is a Dirac delta function, which is an example of a distribution.

References[link]

Further reading[link]

• William E. Boyce and Richard C. DiPrima (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc.. ISBN 0-471-43338-1. 
• Joseph Fourier, translated by Alexander Freeman (published 1822, translated 1878, re-released 2003). The Analytical Theory of Heat. Dover Publications. ISBN 0-486-49531-0.  2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.
• Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly 99 (5): 427–441. DOI:10.2307/2325087. 
• Katznelson, Yitzhak (1976). An introduction to harmonic analysis (Second corrected ed.). New York: Dover Publications, Inc. ISBN 0-486-63331-4. 
• Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert, Springer, Berlin, 1928.
• Walter Rudin (1976). Principles of mathematical analysis (3rd ed.). New York: McGraw-Hill, Inc.. ISBN 0-07-054235-X. 
• A. Zygmund (2002). Trigonometric series (third ed.). Cambridge: Cambridge University Press. ISBN 0-521-89053-5.  The first edition was published in 1935.

External links[link]

• thefouriertransform.com Fourier Series as a prelude to the Fourier Transform
• [2]-Characterizations of a linear subspace associated with Fourier series
• An interactive flash tutorial for the Fourier Series
• Phasor Phactory Allows custom control of the harmonic amplitudes for arbitrary terms
• Java applet shows Fourier series expansion of an arbitrary function
• Example problems - Examples of computing Fourier Series
• Weisstein, Eric W., "Fourier Series" from MathWorld.
• Fourier Series Module by John H. Mathews
• Joseph Fourier - A site on Fourier's life which was used for the historical section of this article
• SFU.ca - 'Fourier Theorem'

This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This biographical article needs additional citations for verification. Please help by adding reliable sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful. (May 2010)

The Basics
The Basics at the Oxford Art Factory, 2010
Background information
Origin	Melbourne, Australia
Genres	Rock
Years active	2002–present
Labels	Independent, Albert Records
Associated acts	Gotye, Down Hills Home, Dog With Wheels, Blood Red Bird
Website	The Basics Official Website
Members
Wally De Backer Kris Schroeder Tim Heath
Past members
Michael Hubbard
Notable instruments
Lead guitar, drums, bass guitar.

The Basics ( /ˈbæzɪtʃ/ BAZZ-ich) are a three-piece band from Melbourne, Australia, formed by Wally De Backer and Kris Schroeder in 2002. Their style has been described as anything from 'indie-pop' to 'rock'n'roll' to 'pop-rock', though their records show they span a wide range of genres, including reggae, ska, country, and electro-pop. They are "recognised as one of Australia's hardest-working bands".^[1]

With close to 1000 shows in their ten-year career, their live performances are well-known for their sense of humour and energy.^[2]

Formation[link]

The Basics were formed after Kris met Wally at a party in Melbourne.^[3] The event was to see off the young aspiring producer, who was about to depart for Los Angeles, and together with a number of musician friends, the two jammed on some blues-rock standards. Later that night, they bonded over a mutual love of The Beatles, 70s and 80s cartoon theme songs and old Sierra adventure games.

Initially, they played around Melbourne as an acoustic guitar/drums combo, starting at The Opposition in Frankston and the House of Fools in Footscray, where in late 2002 they met Michael Hubbard and later invited him to join them on electric guitar.^[4] Though lacking any real experience with the instrument, Kris willingly purchased his first bass guitar, and the trio was born. During this time, the group start performing songs with more complicated three-part harmony.

During late 2002 and early 2003, The Basics recorded and released an album called Get Back through MGM Distribution. They began what was to become regular tours of the East and West coasts of Australia.

2004-2005: Current lineup[link]

Tim Heath joined the group in June 2004, after Michael decided to pursue his solo career, which has included being one half of Down Hills Home.^[4] Tim was reportedly the only person to reply to an advertisement in Melbourne's Beat Magazine looking for a new guitarist. Tim apparently answered the advertisement - which said the applicant "must have a love of 50s and 60s rock'n'roll" - thinking that he would be meeting "a couple of old guys to play with on the weekends over some beers". They soon recorded For Girls Like You.

The Basics in 2005 continued to tour and play small festivals, and after releasing For Girls Like You through MGM Distribution, they departed for a national tour of Australia, which lasted for two months and took them to every Australian state and territory. Stuart Padbury, a young Melbourne sound engineer, joined them on much of this tour. On returning, the band began demonstrating songs for a new album.

2006-2009: Touring and soundtracks[link]

In 2006, The Basics began recording what was to become known as Stand Out/Fit In. Taking several months, it was not completed until the end of the year, despite intentions to have it released in June. Instead, they held a successful 'album preview', where each of the tracks was demonstrated in 'rough form' accompanied by a music video for each (produced by Melbourne photographic artist James Bryans). The band also talked about the origins of the songs. This event brought a lot of attention to the band for the idea's originality. They pre-released a radio single 'Just Hold On' from the album. The band were invited to play the hugely popular Homebake Festival in Sydney, where they received a great response, after being introduced onto the stage by a short stocky Asian guy with big calves ("one of the highlights" wrote The Daily Telegraph).

In 2007, the band departed on a tri-state residency which saw them play Melbourne, Sydney and Brisbane every week for the month of April - a bold move which brought further attention to the band for its original idea.^[5] Riding on the wave of this tour, Schroeder, De Backer and Heath departed on their first overseas tour which saw them performing 26 shows around Japan, Norway and the United Kingdom.^[6] They also toured twice more up the East Coast, one of which was with popular Japanese band The Bawdies. Their song 'Rattle My Chain' was used in a series of commercials for Volvo Australia. The song 'Hey There!' featured in the British film The Bank Job - starring Jason Statham - and was used as the backing for a February 2007 short film directed by Tim Longhurst called The Rip. Their song 'Better' was featured in a surfing documentary about the invention of the shortboard in 1967.

In 2008, the band appeared on Australia's Channel 9 for Australia Day celebrations, performing two of their songs - 'Just Hold On' and 'Hey There!' - to an estimated audience of 2 million. Securing an Australian Government grant, they embarked on a two-month tour of Australia, specifically targeting "culturally underprivileged groups" in rural and regional schoolchildren and Indigenous communities. The programme had them holding masterclasses and performing alongside Indigenous groups in the remote Northern Territory and Queensland.^[7][8] Their initiative also had them raising money for the charity Lifeline^{[disambiguation needed ]}, their contribution to which was recognised with a plaque at the end of the tour. Season two of Californication starring David Duchovny featured their cover of the classic 'Have Love, Will Travel', most famously covered by 'The Sonics'.

In 2009, The Basics for the first time enjoyed the support of Australian radio network Triple J, which had reportedly snubbed the band's previous material.^[9] The singles 'With This Ship' and 'Like A Brother' were added to the station's playlist, and momentum carried The Basics overseas for a second tour of the United Kingdom and Norway, and also a series of shows in Dublin, Ireland. They were joined by the longtime front-of-house engineer Stuart Padbury, who had been mixing them since 2005. The band was often seen in the Grafton Street Mall in Dublin, busking to promote their concerts there. Their second show in Oslo, Norway, saw the band perform with Hot Tub, one of the members of the Norwegian boy band "Boyzvoice", in his first performance for several years. Season Eight of Scrubs featured the song 'Lookin' Over My Shoulder'.

Despite all this promise, the band's album Keep Your Friends Close - produced and part-recorded by Peter Cobbin at London's Abbey Road Studios - though enjoying many favourable reviews,^{[10][11][12][13]} failed to impress Richard Kingsmill, music director of Triple J. This effectively ended the band's apparent upward spiral and the resulting album launch tour was reportedly "trying".

2010: Hiatus and live album[link]

In 2010, following the band's fourth stint at the Tamworth Country Music Festival, and the recording of a yet-unreleased live album, an indefinite hiatus was announced on the band's websites. This was in part due to the impending release of a new album produced by Wally for his Gotye project, and also to reported tensions within the band following the previous year's disappointments.

However, this hiatus was soon broken with a two-month residency at the Northcote Social Club in Melbourne, which saw an apparent return to form.^[14] The band's EP "Wait For You" enjoys airplay on Triple J.^[15] The Basics also played a one-off show at Sydney's Oxford Art Factory.^[16]

On 20 August 2010, The Basics released their fourth LP /ðəˈbæzɪtʃ/, available free and exclusively from their website for digital download^[17]. The band had reportedly moved to expand their listenership through this free giveaway, though 300 "Deluxe" CD/vinyl versions have been printed. /ðəˈbæzɪtʃ/ was recorded at the Northcote Social Club on 6 February, "in front of a live studio audience", and like the previous record, was mixed by Peter Cobbin at Abbey Road Studios. Technically a self-titled album, the phonetics spell out the accurate pronunciation of the band’s name. Kris Schroeder: "For years we’ve worked at bringing our live energy into the studio and thus far it’s been a bit hit-and-miss. Our trick this time was to pull a switcheroo and bring the studio to us instead; the result is the best and closest to us we’ve sounded yet."

Melbourne thespian and Dog With Wheels bassist David Bramble - friends with Heath from university days - has occasionally joined the band on tour, playing keys, often while sporting a large moustache. Other guests have included Jake Mason (saxophone), also of Cookin' on 3 Burners and The Bamboos^{[disambiguation needed ]}, Gideon Brazil (saxophone, flute), Simon Imrei (guitar and vocals) and Monty MacKenzie (saxophone).

Covers have often made an appearance from various sources: to date, some of the bands covered have been Cream, The Kinks, The Rolling Stones, The Police, Neil Young, Bob Dylan, New Kids on the Block, Richard Berry, Sonny Curtis and The Crickets, JJ Cale, Harry Nilsson, Bill Scott, Ryan Adams and The Coasters (via The Beatles for their cover of Three Cool Cats).

Members[link]

Current

• Wally De Backer - drums, vocals (2002-present)
• Kris Schroeder - bass, vocals (2002-present)
• Tim Heath - guitar, vocals (2004-present)

Former

• Michael Hubbard - guitar, vocals (2002-2004)

Discography[link]

Albums[link]

• Get Back (15 January 2004)
• Stand Out/Fit In (5 May 2007)
• Keep Your Friends Close (25 September 2009)
• /ðəˈbæzɪtʃ/ (20 August 2010)

EP[link]

• For Girls Like You (11 July 2005)
• Lookin' Over My Shoulder (10 February 2007)
• Like A Brother (5 June 2009)
• Wait For You (21 May 2010)

Singles[link]

• Call It Rhythm & Blues (20 August 2004)
• Just Hold On
• With This Ship (22 February 2009)

Upcoming projects[link]

The band is taking a three year hiatus, with an announced return to the stage together in early 2014. Wally De Backer launched Making Mirrors under his Gotye act in August 2011, Tim Heath will be touring and recording with Blood Red Bird, and Kris Schroeder recorded a solo EP titled, Patience in the Face of Control, before leaving Australia to work with the Red Cross in Kenya for a two year sabbatical.

References[link]

External links[link]

• Official Website
• Myspace Website