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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.
Josiah Willard Gibbs Born (1839-02-11) February 11, 1839 New Haven, Connecticut, U.S. Died April 28, 1903 (1903-04-28) (aged 64) New Haven, Connecticut, U.S. Nationality American Alma mater Yale College (BA, PhD) Known for List Statistical mechanics Chemical thermodynamics Chemical potential Cross product Dyadics Exergy Principle of maximum work Phase rule Phase space Physical optics Physics ...
All basis functions have nodes at the nodes of the sawtooth, but all but the fundamental have additional nodes. The oscillation about the sawtooth is called the Gibbs phenomenon. 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.
The sinc function, the impulse response for an ideal low-pass filter, illustrating ringing for an impulse. The Gibbs phenomenon, illustrating ringing for a step function.. By definition, ringing occurs when a non-oscillating input yields an oscillating output: formally, when an input signal which is monotonic on an interval has output response which is not monotonic.
The term () is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the sinc {\displaystyle \operatorname {sinc} } function to rolloff the higher frequency Fourier Series coefficients.
Henry Wilbraham (25 July 1825 – 13 February 1883) was an English mathematician. He is known for discovering and explaining the Gibbs phenomenon nearly fifty years before J. Willard Gibbs did. Gibbs and Maxime Bôcher, as well as nearly everyone else, were unaware of Wilbraham's paper on the Gibbs phenomenon.
The Gibbs paradox is a special case of the "mixing paradox" which contains all the salient features. The difference is that the mixing paradox deals with arbitrary distinctions in the two gases, not just distinctions in particle ordering as Gibbs had considered. In this sense, it is a straightforward generalization to the argument laid out by ...
The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an interface between two phases due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may be called thermo-capillary convection [ 1 ] or Bénard–Marangoni convection .