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Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties: [14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity x n; (3) satisfying the boundary condition f (a) = 0; and (4) having zero derivative almost everywhere.
At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true ...
The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.
The points of X where ƒ fails to be a cover are the ramification points of ƒ, and the image of a ramification point under ƒ is called a branch point. For any point P ∈ X and Q = ƒ(P) ∈ Y, there are holomorphic local coordinates z for X near P and w for Y near Q in terms of which the function ƒ(z) is given by =
Since the Gibbs phenomenon comes from undershooting, it may be eliminated by using kernels that are never negative, such as the Fejér kernel. [12] [13]In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation.
There are many known sufficient conditions for the Fourier series of a function to converge at a given point x, for example if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series converges to the average of the left and right limits ...
See the proofs for continuity and discontinuity above for the construction of appropriate neighbourhoods, where has maxima. f {\displaystyle f} is Riemann integrable on any interval and the integral evaluates to 0 {\displaystyle 0} over any set.
Stated precisely, suppose that f is a real-valued function defined on some open interval containing the point x and suppose further that f is continuous at x. If there exists a positive number r > 0 such that f is weakly increasing on (x − r, x] and weakly decreasing on [x, x + r), then f has a local maximum at x.