View source for Dirichlet–Jordan test

In [[Mathematics|mathematics]], the '''Dirichlet–Jordan test''' gives sufficient conditions for a real-valued, [[Periodic function|periodic function]] ''f'' 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|convergence of Fourier series]].

The original test was established by [[Biography:Peter Gustav Lejeune Dirichlet|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 functions. It was extended in the late 19th century by [[Biography:Camille Jordan|Camille Jordan]] to functions of [[Bounded variation|bounded variation]] (any function of bounded variation is the difference of two increasing functions).<ref>{{citation|author=C. Jordan|title= Cours d'analyse de l'Ecole Polytechnique, t.2, calcul integral|publisher= Gauthier-Villars, Paris, 1894}}</ref><ref>{{citation|journal=The American Mathematical Monthly 
|volume=93|year=1986|issue= 4|title=A Simple Proof of the Dirichlet-Jordan Convergence Test|author=Georges A. Lion}}</ref>

==Dirichlet–Jordan test for Fourier series==
The Dirichlet–Jordan test states<ref>{{citation|author=[[Biography:Antoni Zygmund|Antoni Zygmund]]|title=Trigonometric series|year=1952|publisher=Cambridge University Press|page=57}}</ref> that if a periodic function <math>f(x)</math> is of [[Bounded variation|bounded variation]] on a period, then the Fourier series <math>S_n(f(x))</math> converges, as <math>n\to\infty</math>, at each point of the domain to
<math display="block">\lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>
In particular, if <math>f</math> is continuous at <math>x</math>, then the Fourier series converges to <math>f(x)</math>. Moreover, if <math>f</math> is continuous everywhere, then the convergence is uniform.

Stated in terms of a periodic function of period 2π, the Fourier series coefficients are defined as
<math display="block"> a_k = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-ikx}\, dx,</math>
and the partial sums of the Fourier series are
<math display="block">S_n(f(x)) = \sum_{k=-n}^na_k e^{ikx}</math>

The analogous statement holds irrespective of what the period of ''f'' is, or which version of the [[Fourier series]] is chosen.

There is also a pointwise version of the test:<ref>{{citation|author=R. E. Edwards|title=Fourier series: a modern introduction|publisher=Springer|year=1967|page=156}}.</ref> if <math>f</math> is a periodic function in <math>L^1</math>, and is of bounded variation in a neighborhood of <math>x</math>, then the Fourier series at <math>x</math> converges to the limit as above
<math display="block">\lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>

== 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 [[Biography:Joseph Fourier|Joseph Fourier]].<ref name="Fourier series and Fourier integrals">{{citation|author=[[Biography:Jaak Peetre|Jaak Peetre]]|title=On Fourier's discovery of Fourier series and Fourier integrals|year=2000|url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d72e7ff6baf9008d523a192bab2e3400982389d3}}.</ref>

== Dirichlet conditions in signal processing ==
In [[Signal processing|signal processing]],<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> the test is often retained in the original form due to Dirichlet: a piecewise monotone bounded periodic function <math>f</math> 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 is equivalent to having only finitely many local extrema, i.e., that the function changes its variation only finitely many times.<ref>{{citation|title=Mathematica Tutorial for the Second Course. Part V: Convergence of Fourier Series|author=Vladimir Dobrushkin|url=https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch5/converge.html}}: " A function that satisfies the Dirichlet conditions is also called piecewise monotone."</ref><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.<ref>{{citation|page=46|title=Discourse on Fourier series|author=[[Biography:Cornelius Lanczos|Cornelius Lanczos]]|publisher=SIAM|year=2016}}.</ref>)  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.<ref>{{citation|page=48|title=Discourse on Fourier series|author=[[Biography:Cornelius Lanczos|Cornelius Lanczos]]|publisher=SIAM|year=2016}}.</ref>

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

==References==
{{Reflist}}

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

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

{{Sourceattribution|Dirichlet–Jordan test}}