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In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. It is named after its author Peter Gustav Lejeune Dirichlet , and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function 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).
However Carleson's theorem shows that for a given continuous function the Fourier series converges almost everywhere. It is also possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by [ 9 ]
The convolution of D n (x) with any function f of period 2 π is the nth-degree Fourier series approximation to f, i.e., we have () = () = = ^ (), where ^ = is the k th Fourier coefficient of f. This implies that in order to study convergence of Fourier series it is enough to study properties of the Dirichlet kernel.
A Fourier series (/ ˈ f ʊr i eɪ,-i ər / [1]) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are ...
The memoir pointed out Cauchy's mistake and introduced Dirichlet's test for the convergence of series. It also introduced the Dirichlet function as an example of a function that is not integrable (the definite integral was still a developing topic at the time) and, in the proof of the theorem for the Fourier series, introduced the Dirichlet ...
Dirichlet's theorem may refer to any of several mathematical theorems due to Peter Gustav Lejeune Dirichlet. Dirichlet's theorem on arithmetic progressions; Dirichlet's approximation theorem; Dirichlet's unit theorem; Dirichlet conditions; Dirichlet boundary condition; Dirichlet's principle; Pigeonhole principle, sometimes also called Dirichlet ...
By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet -function and also denoted (,). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in ( Dirichlet 1837 ) to prove the theorem on primes in arithmetic progressions that also bears his name.