Dirichlet–Jordan test

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. It was extended in 1894 by Camille Jordan to functions of bounded variation (any function of bounded variation is the difference of two increasing functions).^[2]^[3]

Dirichlet–Jordan test for Fourier series

The Dirichlet–Jordan test states^[4] 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:^[5] 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.^[6] 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.^[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 $f$ has a convergent Fourier series whose value at each point is the arithmetic mean of its left and right limits. The condition of piecewise montonicity is equivalent to having only finitely many local extrema and finitely many discontinuities, i.e., that the function changes its variation only finitely many times.^[9] Any signal that can be physically produced in a laboratory satisfies these conditions.^[10]

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
^ 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.
^ Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals.
^ Alan V. Oppenheim; Alan S. Willsky; Syed Hamish Nawab (1997). Signals & Systems. Prentice Hall. p. 198. ISBN 9780136511755.
^ Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals.
^ B P Lathi (2000), Signal processing and linear systems, Oxford

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

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

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

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

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

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

[7] Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals.

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

[9] Jaak Peetre (2000), On Fourier's discovery of Fourier series and Fourier integrals.

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

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