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
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. B P Lathi (2000), Signal processing and linear systems, Oxford