Dirichlet–Jordan test: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 16:22, 15 October 2024

In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, 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.

The original test was established by Peter Gustav Lejeune Dirichlet in 1829,^[1] for piecewise monotone functions (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).^[2]^[3]^[4]

Dirichlet–Jordan test for Fourier series

The Dirichlet–Jordan test states^[5] that if a periodic function $f(x)$ is of bounded variation on a period, then the Fourier series $S_{n}f(x)$ converges, as $n\to \infty$ , at each point of the domain to $\lim _{\varepsilon \to 0}{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{2}}.$ In particular, if $f$ is continuous at $x$ , then the Fourier series converges to $f(x)$ . Moreover, if $f$ 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 $a_{k}={\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)e^{-ikx}\,dx,$ and the partial sums of the Fourier series are $S_{n}f(x)=\sum _{k=-n}^{n}a_{k}e^{ikx}$

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:^[6] if $f$ is a periodic function in $L^{1}$ , and is of bounded variation in a neighborhood of $x$ , then the Fourier series at $x$ converges to the limit as above $\lim _{\varepsilon \to 0}{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{2}}.$

Jordan test for Fourier integrals

For the Fourier transform on the real line, there is a version of the test as well.^[7] Suppose that $f(x)$ is in $L^{1}(-\infty ,\infty )$ and of bounded variation in a neighborhood of the point $x$ . Then ${\frac {1}{\pi }}\lim _{M\to \infty }\int _{0}^{M}du\int _{-\infty }^{\infty }f(t)\cos u(x-t)\,dt=\lim _{\varepsilon \to 0}{\frac {f(x+\varepsilon )+f(x-\varepsilon )}{2}}.$ If $f$ 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 $f(x)$ .

This version of the test (although not satisfying modern demands for rigor) is historically prior to Dirichlet, being due to Joseph Fourier.^[2]

Dirichlet conditions in signal processing

In signal processing,^[8] the test is often retained in the original form due to Dirichlet: a piecewise monotone bounded periodic function $f$ (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, i.e., that the function changes its variation only finitely many times. This may be called a function of "finite variation", as opposed to bounded variation.^[2]^[9] Finite variation implies bounded variation, but the reverse is not true. (Dirichlet required in addition that the function have only finitely many discontinuities, but this constraint is unnecessarily stringent.^[10]) Any signal that can be physically produced in a laboratory satisfies these conditions.^[11]

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., $L^{1}$ ) over a period, provided it satisfies the other conditions of the test in a neighborhood of the point $x$ where the limit is taken.^[12]

References

^ Dirichlet (1829), "Sur la convergence des series trigonometriques qui servent à represénter une fonction arbitraire entre des limites donnees", J. Reine Angew. Math., 4: 157–169
^ ^a ^b ^c Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals
^ C. Jordan, Cours d'analyse de l'Ecole Polytechnique, t.2, calcul integral, Gauthier-Villars, Paris, 1894
^ Georges A. Lion (1986), "A Simple Proof of the Dirichlet-Jordan Convergence Test", The American Mathematical Monthly, 93 (4)
^ Antoni Zygmund (1952), Trigonometric series, Cambridge University Press, p. 57
^ R. E. Edwards (1967), Fourier series: a modern introduction, Springer, p. 156.
^ E. C. Titchmarsh (1948), Introduction to the theory of Fourier integrals, Oxford Clarendon Press, p. 13.
^ Alan V. Oppenheim; Alan S. Willsky; Syed Hamish Nawab (1997). Signals & Systems. Prentice Hall. p. 198. ISBN 9780136511755.
^ Vladimir Dobrushkin, Mathematica Tutorial for the Second Course. Part V: Convergence of Fourier Series: " A function that satisfies the Dirichlet conditions is also called piecewise monotone."
^ Cornelius Lanczos (2016), Discourse on Fourier series, SIAM, p. 46.
^ B P Lathi (2000), Signal processing and linear systems, Oxford
^ Cornelius Lanczos (2016), Discourse on Fourier series, SIAM, p. 48.

External links

"Dirichlet conditions". PlanetMath.

[1] Dirichlet (1829), "Sur la convergence des series trigonometriques qui servent à represénter une fonction arbitraire entre des limites donnees", J. Reine Angew. Math., 4: 157–169

[Fourier_series_and_Fourier_integrals-2] Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals

[3] C. Jordan, Cours d'analyse de l'Ecole Polytechnique, t.2, calcul integral, Gauthier-Villars, Paris, 1894

[4] Georges A. Lion (1986), "A Simple Proof of the Dirichlet-Jordan Convergence Test", The American Mathematical Monthly, 93 (4)

[5] Antoni Zygmund (1952), Trigonometric series, Cambridge University Press, p. 57

[6] R. E. Edwards (1967), Fourier series: a modern introduction, Springer, p. 156.

[7] E. C. Titchmarsh (1948), Introduction to the theory of Fourier integrals, Oxford Clarendon Press, p. 13.

[sands-8] Alan V. Oppenheim; Alan S. Willsky; Syed Hamish Nawab (1997). Signals & Systems. Prentice Hall. p. 198. ISBN 9780136511755.

[9] Vladimir Dobrushkin, Mathematica Tutorial for the Second Course. Part V: Convergence of Fourier Series: " A function that satisfies the Dirichlet conditions is also called piecewise monotone."

[10] Cornelius Lanczos (2016), Discourse on Fourier series, SIAM, p. 46.

[11] B P Lathi (2000), Signal processing and linear systems, Oxford

[12] Cornelius Lanczos (2016), Discourse on Fourier series, SIAM, p. 48.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

@@ Line 1: / Line 1: @@
-{{distinguish|Dirichlet boundary condition}}
+{{redirect-distinguish|Dirichlet conditions|Dirichlet boundary condition}}
-In [[mathematics]], the '''Dirichlet conditions''' are [[sufficient condition]] for a real-valued, [[periodic function]] ''f''(''x'') to be equal the sum of its [[Fourier series]] at each point where ''f'' is continuous. Moreover, the behavior at points of discontinuity is determined as well. These conditions are named after [[Johann Peter Gustav Lejeune Dirichlet]].
+In [[mathematics]], the '''Dirichlet–Jordan test''' gives [[sufficient condition]]s for a [[real numbers|real]]-valued, [[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]].
+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><ref>{{citation|journal=The American Mathematical Monthly
-The conditions are:
+|volume=93|year=1986|issue= 4|title=A Simple Proof of the Dirichlet-Jordan Convergence Test|author=Georges A. Lion}}</ref>
-*''f''(''x'') must have a finite number of [[Maxima_and_minima|extrema]] in any given interval
-*''f''(''x'') must have a finite number of [[Classification_of_discontinuities|discontinuities]] in any given interval
-*''f''(''x'') must be [[absolutely integrable]] over a period.
-==Dirichlet's Theorem for 1-Dimensional Fourier Series==
+==Dirichlet–Jordan test for Fourier series==
+The Dirichlet–Jordan test states<ref>{{citation|author=[[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]] on a period, then the Fourier series <math>S_nf(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.
-We state Dirichlet's theorem assuming ''f'' is a periodic function of period 2π with Fourier series expansion
+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_nf(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.
-:<math> f(x) \sim \sum_{n = -\infty}^\infty a_n e^{inx} </math>,
+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
-where
+<math display="block">\lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>
+== Jordan test for Fourier integrals ==
-:<math> a_n = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x) e^{-inx}\, dx. </math>
+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>
-The analogous statement holds irrespective of what the period of ''f'' is, or which version of the Fourier expansion is chosen.
+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>.
-<br>
-:'''Dirichlet's theorem:''' If ''f'' satisfies Dirichlet conditions, then for all ''x'', we have that the series obtained by plugging ''x'' into the Fourier series is convergent, and is given by
-::<math> \sum_{n = -\infty}^\infty a_n e^{inx} = \frac{1}{2}(f(x+) + f(x-)) </math>,
-:where the notation
-::<math> f(x+) = \lim_{y \to x^+} f(y) </math>
-::<math> f(x-) = \lim_{y \to x^-} f(y) </math>
-:denotes the right/left limits of ''f''.
-<br>
-A function satisfying Dirichlet's conditions must have right and left limits at each point of discontinuity, or else the function would need to oscillate at that point, violating the condition on maxima/minima. Note that at any point where ''f'' is continuous,
-:<math> \displaystyle f(x+) = f(x-) = f(x) </math>
-so
-:<math> \frac{1}{2}(f(x+) + f(x-)) = f(x) </math>.
-Thus Dirichlet's theorem says in particular that the Fourier series for ''f'' converges and is equal to ''f'' wherever ''f'' is continuous.
+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]],<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> (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, i.e., that the function changes its variation only finitely many times. This may be called a function of "finite variation", as opposed to bounded variation.<ref name="Fourier series and Fourier integrals"/><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> Finite variation implies bounded variation, but the reverse is not true. (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=[[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=[[Cornelius Lanczos]]|publisher=SIAM|year=2016}}.</ref>
+==See also==
+* [[Dini test]]
+==References==
+{{Reflist}}
 ==External links==
-*{{planetmath reference|id=3891|title=Dirichlet conditions}}
+*{{planetmath reference|urlname=DirichletConditions|title=Dirichlet conditions}}
 [[Category:Fourier series]]
+[[Category:Theorems in analysis]]
+{{DEFAULTSORT:Dirichlet-Jordan test}}
-[[pl:Warunki Dirichleta]]