Dirichlet conditions
In mathematics, the Dirichlet conditions are sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at each point where f is continuous. 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). These conditions are named after Peter Gustav Lejeune Dirichlet.
Dirichlet conditions for Fourier series
A set of Dirichlet conditions, for the covergence of Fourier series of a periodic function [math]\displaystyle{ f }[/math], are:[1][2]
- Function [math]\displaystyle{ f }[/math] is absolutely integrable over a period.
- Function [math]\displaystyle{ f }[/math] has bounded variation over one time period. The functions with bounded variations can have (i) at most a countably infinite number of maxima and minima, and (ii) at most a countably infinite number of finite discontinuities.
Dirichlet's theorem for 1-dimensional Fourier series
We state Dirichlet's theorem assuming f is a periodic function of period 2π with Fourier series coefficients [math]\displaystyle{ a_n = \frac{1}{2\pi} \int_{-\pi}^\pi f(x) e^{-inx}\, dx. }[/math]
The analogous statement holds irrespective of what the period of f is, or which version of the Fourier series is chosen.
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]\displaystyle{ \sum_{n = -\infty}^\infty a_n e^{inx} = \frac{f(x^+) + f(x^-)}{2}, }[/math] where the notation [math]\displaystyle{ f(x^+) = \lim_{y \to x^+} f(y) }[/math] [math]\displaystyle{ f(x^-) = \lim_{y \to x^-} f(y) }[/math] denotes the right/left limits of f.[1]
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{ \frac{f(x^+) + f(x^-)}{2} = f(x). }[/math]
Thus Dirichlet's theorem says in particular that under the Dirichlet conditions the Fourier series for f converges to f(x) wherever f is continuous.
Dirichlet conditions for Fourier transform
If the period of a periodic signal tends to infinity then Fourier series becomes Fourier transform. Fourier transforms of periodic (e.g., sine and cosine) functions also exist in the distributional sense which can be expressed using the Dirac delta function. A set of Dirichlet conditions, for the covergence of Fourier transform of an aperiodic function [math]\displaystyle{ g }[/math], are:[1][2]
- Function [math]\displaystyle{ g }[/math] is absolutely integrable over the entire duration of time.
- Function [math]\displaystyle{ g }[/math] has bounded variation over the entire duration of time. The functions with bounded variations can contain (i) at most a countably infinite number of maxima and minima, and (ii) at most a countably infinite number of finite discontinuities.
References
- ↑ 1.0 1.1 1.2 Alan V. Oppenheim; Alan S. Willsky; Syed Hamish Nawab (1997). Signals & Systems. Prentice Hall. p. 198. ISBN 9780136511755. https://books.google.com/books?id=O9ZHSAAACAAJ&q=signals+and+systems.
- ↑ 2.0 2.1 Singh, Pushpendra; Singhal, Amit; Fatimah, Binish; Gupta, Anubha; Joshi, Shiv Dutt (September 2022). "Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes]". IEEE Signal Processing Magazine 39 (5): 77–84. doi:10.1109/MSP.2022.3172620.
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