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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).
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.
There exist continuous functions whose Fourier series converges pointwise but not uniformly. [8] However, the Fourier series of a continuous function need not converge pointwise. Perhaps the easiest proof uses the non-boundedness of Dirichlet's kernel in L 1 (T) and the Banach–Steinhaus uniform boundedness principle.
The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem. [39] [40] [41] [42]
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 ...
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.
This is the analogue for Dirichlet series of the radius of convergence for power series. The Dirichlet series case is more complicated, though: absolute convergence and uniform convergence may occur in distinct half-planes. In many cases, the analytic function associated with a Dirichlet series has an analytic extension to a larger domain.
1.11 Dirichlet's test. 1.12 Cauchy's convergence test. 1.13 Stolz–Cesàro theorem. ... for instance for Fourier series there is the Dini test.