Editing Dirichlet–Jordan test

{{redirect-distinguish|Dirichlet conditions|Dirichlet boundary condition}}
In [[mathematics]], the '''Dirichlet–Jordan test''' gives [[sufficient condition]]s for a [[complex numbers|complex-valued]], [[periodic function]] <math>f</math> to be equal to the sum of its [[Fourier series]] at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). It is one of many conditions for the [[convergence of Fourier series]].

The original test was established by [[Peter Gustav Lejeune Dirichlet]] in 1829,<ref>{{citation|author=Dirichlet|year=1829|title=Sur la convergence des series trigonometriques qui servent à represénter une fonction arbitraire entre des limites donnees|journal=J. Reine Angew. Math.|volume= 4|pages=157–169}}</ref> for piecewise [[monotone function]]s (functions with a finite number of sections per period each of which is monotonic). It was extended in the late 19th century by [[Camille Jordan]] to functions of [[bounded variation]] in each period (any function of bounded variation is the difference of two monotonically increasing functions).<ref name="Fourier series and Fourier integrals"/><ref>{{citation|author=C. Jordan|title= Cours d'analyse de l'Ecole Polytechnique, t.2, calcul integral|publisher= Gauthier-Villars, Paris, 1894}}</ref>

==Dirichlet–Jordan test for Fourier series==
Let <math>f(x)</math> be complex-valued [[Lebesgue_integral#Complex-valued_functions|integrable]] function on the interval <math>[-\pi,\pi]</math> and the [[Series_(mathematics)#Partial_sum_of_a_series|partial sums]] of its Fourier series <math>S_nf(x)</math>, given by
<math display="block">S_nf(x) = \sum_{k=-n}^nc_k e^{ikx},</math>
with [[Fourier coefficients]] <math>c_k</math> defined as
<math display="block"> c_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-ikx}\, dx.</math>
The Dirichlet-Jordan test states that if <math>f</math> is of [[bounded variation]], then for each <math>x \in [-\pi,\pi]</math> the limit <math>S_nf(x)</math> exists and is equal to{{sfn|Zygmund|Fefferman|2003|p=57}}{{sfn|Lion|1986|pp=281–282}}
<math display="block">\lim_{n \to \infty} S_nf(x) =\lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>
Alternatively, Jordan's test states that if <math>f\in L^1</math> is of bounded variation in a neighborhood of <math>x</math>, then the limit of <math>S_nf(x)</math> exists and converges in a similar manner.{{sfn|Edwards|1979|p=156}}  

If, in addition, <math>f</math> is continuous at <math>x</math>, then
<math display="block">\lim_{n \to \infty} S_nf(x) = f(x).</math>
Moreover, if <math>f</math> is continuous at every point in <math>[-\pi,\pi]</math>, then the convergence is [[uniform convergence|uniform]] rather than just [[pointwise convergence|pointwise]].

The analogous statement holds irrespective of the choice of period of <math>f</math>, or which [[Fourier_series#Synthesis|version of the Fourier series]] is chosen.

== Jordan test for Fourier integrals ==
For the [[Fourier transform]] on the real line, there is a version of the test as well.<ref>{{citation|author=[[E. C. Titchmarsh]]|title=Introduction to the theory of Fourier integrals|year=1948|page=13|publisher=Oxford Clarendon Press}}.</ref>  Suppose that <math>f(x)</math> is in <math>L^1(-\infty,\infty)</math> and of bounded variation in a neighborhood of the point <math>x</math>.  Then
<math display="block">\frac1\pi\lim_{M\to\infty}\int_0^{M}du\int_{-\infty}^\infty f(t)\cos u(x-t)\,dt = \lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>
If <math>f</math> is continuous in an open interval, then the integral on the left-hand side converges uniformly in the interval, and the limit on the right-hand side is <math>f(x)</math>.

This version of the test (although not satisfying modern demands for rigor) is historically prior to Dirichlet, being due to [[Joseph Fourier]].<ref name="Fourier series and Fourier integrals">{{citation|author=[[Jaak Peetre]]|title=On Fourier's discovery of Fourier series and Fourier integrals|year=2000|url=https://web.archive.org/web/20221201121132/https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d72e7ff6baf9008d523a192bab2e3400982389d3}}</ref>

== Dirichlet conditions in signal processing ==
In [[signal processing]], the test is often retained in the original form due to Dirichlet:<ref name='sands'>{{cite book|last1= Alan V. Oppenheim|last2= Alan S. Willsky|last3= Syed Hamish Nawab|year= 1997|title= Signals & Systems|url= https://books.google.com/books?id=O9ZHSAAACAAJ&q=signals+and+systems|publisher= Prentice Hall| isbn= 9780136511755|page= 198}}</ref>{{sfn|Khare|Butola|Rajora|2023|p=9}}{{sfn|Proakis|Manolakis|1996|p=234}} a piecewise monotone bounded periodic function <math>f</math> (having a finite number of monotonic intervals per period) has a convergent Fourier series whose value at each point is the arithmetic mean of the left and right limits of the function.  The condition of piecewise monotonicity stipulates having only finitely many local extrema per period, which implies <math>f</math> is of bounded variation (though the reverse is not true).<ref name="Fourier series and Fourier integrals"/> (Dirichlet required in addition that the function have only finitely many discontinuities, but this constraint is unnecessarily stringent.{{sfn|Lanczos|2016|p=46}})  Any signal that can be physically produced in a laboratory satisfies these conditions.<ref>{{citation|author=B P Lathi|title=Signal processing and linear systems|year=2000|publisher=Oxford}}</ref>

As in the pointwise case of the Jordan test, the condition of boundedness can be relaxed if the function is assumed to be [[absolutely integrable]] (i.e., <math>L^1</math>) over a period, provided it satisfies the other conditions of the test in a neighborhood of the point <math>x</math> where the limit is taken.{{sfn|Lanczos|2016|p=48}}

==See also==
* [[Dini test]]

==Notes==
{{Reflist}}
==References==
* {{cite book | last=Edwards | first=R. E. | title=Fourier Series | publisher=Springer New York | publication-place=New York, NY | volume=64 | date=1979 | isbn=978-1-4612-6210-7 | doi=10.1007/978-1-4612-6208-4}}
* {{cite book | last=Lanczos | first=Cornelius | title=Discourse on Fourier Series | publisher=Society for Industrial and Applied Mathematics | publication-place=Philadelphia, PA | date=2016-09-12 | isbn=978-1-61197-451-5 | doi=10.1137/1.9781611974522 | doi-access=free | url=https://epubs.siam.org/doi/pdf/10.1137/1.9781611974522.fm | access-date=2024-12-15}}
* {{cite journal | last=Lion | first=Georges A. | title=A Simple Proof of the Dirichlet-Jordan Convergence Test | journal=The American Mathematical Monthly | volume=93 | issue=4 | date=1986 | issn=0002-9890 | doi=10.1080/00029890.1986.11971805 | pages=281–282}}
* {{cite book | last1=Khare | first1=Kedar | last2=Butola | first2=Mansi | last3=Rajora | first3=Sunaina | title=Fourier Optics and Computational Imaging | publisher=Springer International Publishing | publication-place=Cham | date=2023 | isbn=978-3-031-18352-2 | doi=10.1007/978-3-031-18353-9}}
* {{cite book|last1=Proakis|first1=John G. |last2=Manolakis|first2=Dimitris G.|author2-link= Dimitris Manolakis |title=Digital Signal Processing: Principles, Algorithms, and Applications|url=https://archive.org/details/digitalsignalpro00proa|url-access=registration|year=1996|publisher=Prentice Hall|isbn=978-0-13-373762-2|edition=3rd}}
* {{cite book | last=Zygmund | first=A. | last2=Fefferman | first2=Robert | title=Trigonometric Series | publisher=Cambridge University Press | date=2003-02-06 | isbn=978-0-521-89053-3 | doi=10.1017/cbo9781316036587}}

==External links==
*{{planetmath reference|urlname=DirichletConditions|title=Dirichlet conditions}}

[[Category:Fourier series]]
[[Category:Theorems in mathematical analysis]]
{{DEFAULTSORT:Dirichlet-Jordan test}}