# Gibbs phenomenon

Gibbs' PhenomenonGibbs artifact
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.wikipedia
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### Ringing artifacts

ringingringing artifactfilter overshoots
This is one cause of ringing artifacts in signal processing. From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts.
Mathematically, this is called the Gibbs phenomenon.

### Henry Wilbraham

WilbrahamWilbraham, Henry
:is sometimes known as the Wilbraham–Gibbs constant.
He is known for discovering and explaining the Gibbs phenomenon nearly fifty years before J. Willard Gibbs did.

### Albert A. Michelson

Albert Abraham MichelsonAlbert MichelsonMichelson
In 1898, Albert A. Michelson developed a device that could compute and re-synthesize the Fourier series.
In 1898, he noted the Gibbs phenomenon in Fourier analysis on a mechanical computer that was constructed by him.

### Josiah Willard Gibbs

Willard GibbsJ. Willard GibbsGibbs
Inspired by some correspondence in Nature between Michelson and Love about the convergence of the Fourier series of the square wave function, in 1898 J. Willard Gibbs published a short note in which he considered what today would be called a sawtooth wave and pointed out the important distinction between the limit of the graphs of the partial sums of the Fourier series, and the graph of the function that is the limit of those partial sums.
In other mathematical work, he re-discovered the "Gibbs phenomenon" in the theory of Fourier series (which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, Henry Wilbraham).

### Square wave

squaresquare-wavepulse
The three pictures on the right demonstrate the phenomenon for a square wave (of height \pi/4) whose Fourier expansion is
A curiosity of the convergence of the Fourier series representation of the square wave is the Gibbs phenomenon.

### Convergence of Fourier series

classic harmonic analysisdivergesmore about absolute convergence of Fourier series
See more about absolute convergence of Fourier series.
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 (but see Gibbs phenomenon).

### Fourier series

Fourier coefficientFourier expansionFourier coefficients
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

### Low-pass filter

low-passlow pass filterlowpass filter
From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts.
An ideal low-pass filter results in ringing artifacts via the Gibbs phenomenon.

### Wavelet

waveletswavelet analysisscaling function
Using a continuous wavelet transform, the wavelet Gibbs phenomenon never exceeds the Fourier Gibbs phenomenon.
However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as Gibbs phenomenon.

### Ringing (signal)

ringingpre-ringingring
From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts.
Ringing artifacts are also present in square waves; see Gibbs phenomenon.

### Fejér kernel

Fejer meanFejérFejér summation
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.

### Runge's phenomenon

oscillations between exact-fit valuesRunge's function
The phenomenon is similar to the Gibbs phenomenon in Fourier series approximations.

### Trigonometric integral

sine integralCosine integralHyperbolic cosine integral
In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the sine integral; for a square wave the description is not as simply stated.
Related is the Gibbs phenomenon: if the sine integral is considered as the convolution of the sinc function with the heaviside step function, this corresponds to truncating the Fourier series, which is the cause of the Gibbs phenomenon.

### Pinsky phenomenon

This lack of convergence in the Pinsky phenomenon happens far away from the boundary of the discontinuity, rather than at the discontinuity itself seen in the Gibbs phenomenon.

### Sinc function

sinccardinal sine functioncardinal sine
This can be represented as convolution of the original signal with the impulse response of the filter (also known as the kernel), which is the sinc function.
A similar situation is found in the Gibbs phenomenon.

### Impulse response

impulseImpulse Response Functionsimpulse-response function.
This can be represented as convolution of the original signal with the impulse response of the filter (also known as the kernel), which is the sinc function.

### Sigma approximation

Lanczos sigma factorσ-approximationsigma-approximation
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.
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.

### Mach bands

Mach bandMach bandingMach band illusion

### Mathematics

mathematicalmathmathematician
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

### Piecewise

piecewise continuouspiecewise functionpiecewise smooth
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

### Periodic function

periodicperiodperiodicity
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

### Classification of discontinuities

discontinuitiesdiscontinuousdiscontinuity
In mathematics, the Gibbs phenomenon, discovered by and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

### Series (mathematics)

infinite seriesseriespartial sum
The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself.

### Signal processing

signal analysissignalsignal processor
This is one cause of ringing artifacts in signal processing. From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts.

### Integer

integersintegralZ
More precisely, this is the function f which equals \pi/4 between 2n\pi and (2n+1)\pi and -\pi/4 between (2n+1)\pi and (2n+2)\pi for every integer n; thus this square wave has a jump discontinuity of height \pi/2 at every integer multiple of \pi.