The Gibbs phenomenon is a phenomenon in mathematics that occurs when approximating a function that has a jump discontinuity or a point of discontinuity with a Fourier series. The Fourier series is a mathematical technique that represents a periodic function as a sum of sinusoidal functions. The Gibbs phenomenon is the overshoot of the Fourier series at a discontinuity, where the overshoot does not decrease with increasing the number of terms in the series.

The Gibbs phenomenon can be observed when a Fourier series is used to approximate a step function, such as the Heaviside step function, which is defined as zero for negative values and one for positive values. The Fourier series approximation of this function is a series of sine functions that oscillate around the step discontinuity, with the oscillations increasing as the number of terms in the series increases.

The Gibbs phenomenon occurs because the Fourier series tries to fit the function by minimizing the overall error, but cannot avoid the overshoot at the point of discontinuity. This overshoot is caused by the high-frequency components of the Fourier series, which become more significant as the number of terms in the series increases.

The Gibbs phenomenon has important implications in many fields, including signal processing and image processing, where it can lead to visual artifacts and distortion in reconstructed signals. There are various methods for mitigating the Gibbs phenomenon, including windowing functions and the use of alternative approximation techniques, such as wavelets. Despite this, the Gibbs phenomenon remains a fundamental limitation of the Fourier series and is an important topic of study in mathematics and engineering.
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