In
mathematics, the Dirichlet–Jordan test gives
sufficient conditions for a
real-valued,
periodic functionf 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 Dirichlet–Jordan test states[4] that if a periodic function is of
bounded variation on a period, then the Fourier series converges, as , at each point of the domain to
In particular, if is continuous at , then the Fourier series converges to . Moreover, if 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
and the partial sums of the Fourier series are
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:[5] if is a periodic function in , and is of bounded variation in a neighborhood of , then the Fourier series at converges to the limit as above
Jordan test for Fourier integrals
For the
Fourier transform on the real line, there is a version of the test as well.[6] Suppose that is in and of bounded variation in a neighborhood of the point . Then
If 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 .
This version of the test (although not satisfying modern demands for rigor) is historically prior to Dirichlet, being due to
Joseph Fourier.[7]
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 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.[9][7] (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., ) over a period, provided it satisfies the other conditions of the test in a neighborhood of the point where the limit is taken.[12]
^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
^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.