Фурье қатарларындағы тербеліс қатесі және Гиббс құбылысы

Oscillatory error in Fourier series
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The th partial Fourier series of the function (formed by summing the lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere (pointwise convergence on continuous points) except points of discontinuity. The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus, but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. It is named after Josiah Willard Gibbs.

The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following:As more Fourier series constituents or components are taken, the Fourier series shows the first overshoot in the oscillatory behavior around the jump point approaching ~ 9% of the (full) jump and this oscillation does not disappear but gets closer to the point so that the integral of the oscillation approaches zero. At the jump point, the Fourier series gives the average of the function's both side limits toward the point.

The Gibbs phenomenon was first noticed and analyzed by Henry Wilbraham in an 1848 paper. The paper attracted little attention until 1914 when it was mentioned in Heinrich Burkhardt's review of mathematical analysis in Klein's encyclopedia. In 1898, Albert A. Michelson developed a device that could compute and re synthesize the Fourier series. A widespread myth says that when the Fourier coefficients for a square wave were input to the machine, the graph would oscillate at the discontinuities, and that because it was a physical device subject to manufacturing flaws, Michelson was convinced that the overshoot was caused by errors in the machine. In fact the graphs produced by the machine were not good enough to exhibit the Gibbs phenomenon clearly, and Michelson may not have noticed it as he made no mention of this effect in his paper about his machine or his later letters to Nature. Inspired by correspondence in Nature between Michelson and A. E. H. Love about the convergence of the Fourier series of the square wave function, J. Willard Gibbs published a note in 1898 pointing out the important distinction between the limit of the graphs of the partial sums of the Fourier series of a sawtooth wave and the graph of the limit of those partial sums. In his first letter Gibbs failed to notice the Gibbs phenomenon, and the limit that he described for the graphs of the partial sums was inaccurate. In 1899 he published a correction in which he described the overshoot at the point of discontinuity (Nature, April 27, 1899, p. 606). In 1906, Maxime Bôcher gave a detailed mathematical analysis of that overshoot, coining the term "Gibbs phenomenon" and bringing it into widespread use.

Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sinusoidal waves. It is important to put emphasis on the word finite, because even though every partial sum of the Fourier series overshoots around each discontinuity it is approximating, the limit of summing an infinite number of sinusoidal waves does not. The overshoot peaks moves closer and closer to the discontinuity as more terms are summed, so convergence is possible. There is no contradiction (between the overshoot error converging to a non zero height even though the infinite sum has no overshoot), because the overshoot peaks move toward the discontinuity. The Gibbs phenomenon thus exhibits pointwise convergence, but not uniform convergence. For a piecewise continuously differentiable (class C1) function, the Fourier series converges to the function at every point except at jump discontinuities. At jump discontinuities, the infinite sum will converge to the jump discontinuity's midpoint (i. e. the average of the values of the function on either side of the jump), as a consequence of Dirichlet's theorem. The Gibbs phenomenon is closely related to the principle that the smoothness of a function controls the decay rate of its Fourier coefficients. Fourier coefficients of smoother functions will more rapidly decay (resulting in faster convergence), whereas Fourier coefficients of discontinuous functions will slowly decay (resulting in slower convergence). For example, the discontinuous square wave has Fourier coefficients that decay only at the rate of , while the continuous triangle wave has Fourier coefficients that decay at a much faster rate of
This only provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be uniformly convergent by the Weierstrass M test and would thus be unable to exhibit the above oscillatory behavior. By the same token, it is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction. See .

Since the Gibbs phenomenon comes from undershooting, it may be eliminated by using kernels that are never negative, such as the Fejér kernel. 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. Using a continuous wavelet transform, the wavelet Gibbs phenomenon never exceeds the Fourier Gibbs phenomenon. Also, using the discrete wavelet transform with Haar basis functions, the Gibbs phenomenon does not occur at all in the case of continuous data at jump discontinuities, and is minimal in the discrete case at large change points. In wavelet analysis, this is commonly referred to as the Longo phenomenon. In the polynomial interpolation setting, the Gibbs phenomenon can be mitigated using the S Gibbs algorithm.

The Gibbs phenomenon is undesirable because it causes artifacts, namely clipping from the overshoot and undershoot, and ringing artifacts from the oscillations. In the case of low pass filtering, these can be reduced or eliminated by using different low pass filters. In MRI, the Gibbs phenomenon causes artifacts in the presence of adjacent regions of markedly differing signal intensity. This is most commonly encountered in spinal MRIs where the Gibbs phenomenon may simulate the appearance of syringomyelia. The Gibbs phenomenon manifests as a cross pattern artifact in the discrete Fourier transform of an image, where most images (e. g. micrographs or photographs) have a sharp discontinuity between boundaries at the top / bottom and left / right of an image. When periodic boundary conditions are imposed in the Fourier transform, this jump discontinuity is represented by continuum of frequencies along the axes in reciprocal space (i. e. a cross pattern of intensity in the Fourier transform). And although this article mainly focused on the difficulty with trying to construct discontinuities without artifacts in the time domain with only a partial Fourier series, it is also important to consider that because the inverse Fourier transform is extremely similar to the Fourier transform, there equivalently is difficulty with trying to construct discontinuities in the frequency domain using only a partial Fourier series. Thus for instance because idealized brick wall and rectangular filters have discontinuities in the frequency domain, their exact representation in the time domain necessarily requires an infinitely long sinc filter impulse response, since a finite impulse response will result in Gibbs rippling in the frequency response near cut off frequencies, though this rippling can be reduced by windowing finite impulse response filters (at the expense of wider transition bands).