Editing Fourier series (section)

== Fourier theorem proving convergence of Fourier series ==
{{main|Convergence of Fourier series}}

In [[engineering]], the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are usually better-behaved than those in other disciplines. In particular, if <math>s</math> is continuous and the derivative of <math>s(x)</math> (which may not exist everywhere) is square integrable, then the Fourier series of <math>s</math> converges absolutely and uniformly to <math>s(x)</math>.<ref>{{cite book|last=Tolstov|first=Georgi P.|url=https://books.google.com/books?id=XqqNDQeLfAkC&q=fourier-series+converges+continuous-function&pg=PA82|title=Fourier Series|publisher=Courier-Dover|year=1976|isbn=0-486-63317-9}}</ref> If a function is [[Square-integrable function|square-integrable]] on the interval <math>[x_0,x_0+P]</math>, then the Fourier series [[Carleson's theorem|converges]] to the function [[almost everywhere]]. It is possible to define Fourier coefficients for more general functions or distributions, in which case [[pointwise convergence]] often fails, and convergence in norm or [[Weak convergence (Hilbert space)|weak convergence]] is usually studied.

<gallery widths="224" heights="224">
Fourier_series_square_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/b/bd/Fourier_series_square_wave_circles_animation.svg|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases [{{filepath:Fourier_series_square_wave_circle_animation.svg}} (animation)]
Fourier_series_sawtooth_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/1/1e/Fourier_series_sawtooth_wave_circles_animation.svg|Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases [{{filepath:Fourier_series_sawtooth_wave_circles_animation.svg}} (animation)]
Example_of_Fourier_Convergence.gif |Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" ([[Gibbs phenomenon]]) at the transitions to/from the vertical sections.
</gallery>

The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the [[Dirichlet conditions]]), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as ''Fourier's theorem'' or ''the Fourier theorem''.<ref>{{cite book |last=Siebert |first=William McC. |url=https://books.google.com/books?id=zBTUiIrb2WIC&q=%22fourier%27s+theorem%22&pg=PA402 |title=Circuits, signals, and systems |publisher=MIT Press |year=1985 |isbn=978-0-262-19229-3 |page=402}}</ref><ref>{{cite book |last1=Marton |first1=L. |url=https://books.google.com/books?id=27c1WOjCBX4C&q=%22fourier+theorem%22&pg=PA369 |title=Advances in Electronics and Electron Physics |last2=Marton |first2=Claire |publisher=Academic Press |year=1990 |isbn=978-0-12-014650-5 |page=369}}</ref><ref>{{cite book |last=Kuzmany |first=Hans |url=https://books.google.com/books?id=-laOoZitZS8C&q=%22fourier+theorem%22&pg=PA14 |title=Solid-state spectroscopy |publisher=Springer |year=1998 |isbn=978-3-540-63913-8 |page=14}}</ref><ref>{{cite book |last1=Pribram |first1=Karl H. |url=https://books.google.com/books?id=nsD4L2zsK4kC&q=%22fourier+theorem%22&pg=PA26 |title=Brain and perception |last2=Yasue |first2=Kunio |last3=Jibu |first3=Mari |publisher=Lawrence Erlbaum Associates |year=1991 |isbn=978-0-89859-995-4 |page=26}}</ref>

===Least squares property===
The earlier {{EquationNote|Eq.2}}:
:<math>s_N(x) = \sum_{n=-N}^N S[n]\ e^{i 2\pi\tfrac{n}{P} x},</math>
is a [[trigonometric polynomial]] of degree <math>N</math> that can be generally expressed as''':'''
:<math>p_N(x)=\sum_{n=-N}^N p[n]\ e^{i 2\pi\tfrac{n}{P}x}.</math>
[[Parseval's theorem]] implies that:

{{math theorem | math_statement=The trigonometric polynomial <math>s_N</math> is the unique best trigonometric polynomial of degree <math>N</math> approximating <math>s(x)</math>, in the sense that, for any trigonometric polynomial <math>p_N \neq s_N</math> of degree <math>N</math>, we have:
<math display="block">\|s_N - s\|_2 < \|p_N - s\|_2,</math>
where the Hilbert space norm is defined as:
<math display="block">\| g \|_2 = \sqrt{{1 \over P} \int_P |g(x)|^2 \, dx}.</math>
}}

===Convergence theorems ===
{{See also|Gibbs phenomenon}}
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.

{{math theorem | math_statement= If <math>s</math> belongs to <math>L^2 (P)</math>, then <math>s_N</math> converges to <math>s</math> in <math>L^2 (P)</math> as <math>N \to \infty</math>, that is: <math display="block">\lim_{N\to \infty}\|s_N - s\|_2=0.</math>}}
If <math>s</math> is continuously differentiable, then <math>(i n) S[n]</math> is the <math>n^{\text{th}}</math> Fourier coefficient of the first derivative <math>s'</math>. Since  <math>s'</math> is continuous, and therefore bounded, it is [[square-integrable]] and its Fourier coefficients are square-summable. Then, by the [[Cauchy–Schwarz inequality]],

:<math>\left(\sum_{n\ne 0}|S[n]|\right)^2\le \sum_{n\ne 0}\frac1{n^2}\cdot\sum_{n\ne 0} |nS[n]|^2.</math>

This means that <math>s</math> is [[absolutely summable]]. The sum of this series is a continuous function, equal to <math>s</math>, since the Fourier series converges in <math>L^1</math> to <math>s</math>:

{{math theorem| math_statement= If <math>s \in C^1(\mathbb{R})</math>, then <math>s_N</math> converges to <math>s</math> [[uniform convergence|uniformly]].}}

This result can be proven easily if <math>s</math> is further assumed to be <math>C^2</math>, since in that case <math>n^2S[n]</math> tends to zero as <math>n \rightarrow \infty</math>. More generally, the Fourier series is absolutely summable, thus converges uniformly to <math>s</math>, provided that <math>s</math> satisfies a [[Hölder condition]] of order <math>\alpha > 1/2</math>. In the absolutely summable case, the inequality:
:<math>\sup_x |s(x) - s_N(x)| \le \sum_{|n| > N} |S[n]|</math>

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 <math>x</math> if <math>s</math> is differentiable at <math>x</math>, to more sophisticated results such as [[Carleson's theorem]] which states that the Fourier series of an <math>L^2</math> function converges [[almost everywhere]].

=== Divergence ===
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 titled ''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.{{sfn|Katznelson|2004}}

It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function ''f'' defined for all ''x'' in [0,π] by<ref>{{cite book |last=Gourdon |first= Xavier|title= Les maths en tête. Analyse (2ème édition)|language= french| date=2009 |publisher= Ellipses|page=264 |isbn=978-2729837594}}</ref>

:<math>f(x) = \sum_{n=1}^{\infty} \frac{1}{n^2} \sin\left[ \left( 2^{n^3} +1 \right) \frac{x}{2}\right].</math>

Because the function is even the Fourier series contains only cosines:

:<math>\sum_{m=0}^\infty C_m \cos(mx).</math>

The coefficients are:

:<math>C_m=\frac 1\pi\sum_{n=1}^{\infty} \frac{1}{n^2} \left\{\frac 2{2^{n^3} +1-2m}+\frac 2{2^{n^3} +1+2m}\right\}</math>

As {{mvar|m}} increases, the coefficients will be positive and increasing until they reach a value of about <math>C_m\approx 2/(n^2\pi)</math> at <math>m=2^{n^3}/2</math> for some {{mvar|n}} and then become negative (starting with a value around <math>-2/(n^2\pi)</math>) and getting smaller, before starting a new such wave. At <math>x=0</math> the Fourier series is simply the running sum of <math>C_m,</math> and this builds up to around

:<math>\frac 1{n^2\pi}\sum_{k=0}^{2^{n^3}/2}\frac 2{2k+1}\sim\frac 1{n^2\pi}\ln 2^{n^3}=\frac n\pi\ln 2</math>

in the {{mvar|n}}th wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.