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A Fourier partial sum retains the modes whose integer frequency multipliers satisfy $|k|\le n$ in a complex Fourier series:For a real function of period $2\pi$, this is equivalently $a_0/2+{\sum }_{k=1}^n[a_k\cos (kx)+b_k\sin (kx)]$. It is a trigonometric polynomial, even when the original function is discontinuous. For a square wave, retaining $M$ nonzero odd sine modes gives $S_{2M-1}f$; all even-frequency coefficients are zero. The Gibbs phenomenon illustrates why more retained modes need not reduce the maximum error.

A Fourier series describes a periodic function through sines and cosines whose frequencies are integer multiples of one base frequency. These Fourier harmonics give a useful alternative to describing the value at every input: each Fourier coefficient measures how much of one oscillating pattern is present. For a real, integrable function of period $2\pi$, the real form isThe cosine formula includes $n=0$; the sine formula uses $n\ge 1$. Thus $a_0/2$ is the mean value over a period, not $a_0$ itself. The symbol $\sim$ associates a function with its Fourier coefficients; an equality requires a specified kind of convergence, such as pointwise convergence. For general period $T>0$, replace $nx$ by $2\pi nt/T$ and the coefficient prefactor by $2/T$. Fourier orthogonality explains why integrating against one mode removes the others. ^[ref].

For a finite example, takeIts mean is $1/2$, its real Fourier coefficients are $a_0=1$, $a_1=1$, $b_2=1/2$, and all other $a_n,b_n$ vanish. At $x=\pi /4$, the three contributions add to $1/2+1/\sqrt{2}+1/2=1+1/\sqrt{2}\approx 1.70710678$. The function is already a trigonometric polynomial, so retaining these modes reconstructs it exactly. Here $x$ is an angular coordinate in radians; if it instead represents time, a base angular frequency must supply the corresponding units.

The change of viewpoint is useful because many linear problems act separately on these modes. The second derivative maps $\sin (nx)$ to $-n^2\sin (nx)$ and maps $\cos (nx)$ in the same way. Consequently a differential equation that is complicated for a whole profile can become a simpler calculation for each Fourier harmonic. A linear time-invariant system similarly has a separate response to each complex exponential. Superposition combines the responses when the relevant sums and operations are justified. ^[ref].

The infinite case contains a subtlety absent from the finite example. Under pointwise convergence of a piecewise smooth Fourier series, the value at a jump discontinuity is the average of the two one-sided limits. For a square-integrable function, the complex Fourier series instead converges in the $L^2$ norm: the integrated squared error tends to zero. Neither statement says that the error becomes small everywhere at once. A square wave shows the difference vividly through the Gibbs phenomenon.

A historical source for this connection between profiles and modes is Joseph Fourier's 1822 book Théorie analytique de la chaleur. Chapter III develops expansions in sines and cosines while studying the propagation of heat. The heat equation gives a continuing application of this approach. ^[ref].

The Gibbs phenomenon is the persistent overshoot near a jump discontinuity in ordinary truncated Fourier series. For this square wave, retain $M$ nonzero odd modes:As $M$ grows, the first peak approaches the jump while its height tends to approximately $1.17897974$. This is an excess of about $0.17898$ above the upper level $1$, or $8.949\%$ of the jump size $2$. The narrower region does not imply a vanishing maximum error. At the jump itself every $Q_M(0)=0$. These observations are compatible with pointwise convergence and $L^2$ convergence but exclude uniform convergence to the discontinuous square wave. ^[ref].

For this particular sum, its derivative is $Q_M^{\prime }(x)=(2/\pi )\sin (2Mx)/\sin x$ away from multiples of $\pi$, so the first positive peak is exactly at $x=\pi /(2M)$. There,This is a midpoint Riemann sum tending to $2\mathrm{Si}(\pi )/\pi$, where $\mathrm{Si}$ is the sine integral. It gives a short explanation of the nonzero limiting excess.

The phenomenon matters whenever a sharp boundary is represented by a limited set of Fourier harmonics: a visible ripple can be a property of the approximation rather than an oscillation in the original profile. Josiah Willard Gibbs reported the phenomenon in his 1899 letter “Fourier's Series” in Nature, volume 59, page 606. This is a documented historical report, without a claim of first discovery. ^[ref].

Taking the inner product with $f_m$ and using Fourier orthogonality isolates $a_m$. Since $x\sin (mx)$ is even, integration by parts givesConsequently the Fourier sine series isThe series converges to $x$ at each interior point and in the L2 norm. Its periodic extension has jumps at $\pm \pi$, where the Fourier series instead converges to the midpoint value zero. The overshoot near those jumps is the Gibbs phenomenon.

First, the Fourier coefficients of the Fourier cosine series decay as $O(n^{-2})$, whereas those of the Fourier sine series are of order $n^{-1}$. In particular, the absolute convergence bound for the Fourier cosine series gives a uniform tail estimateThus its tail is absolutely controlled using fewer Fourier harmonics. The Fourier sine series does not have an absolutely summable sequence of Fourier coefficients.

Second, the even extension is continuous, so its Fourier cosine series converges uniformly on the closed interval and approximates the nonzero endpoint limits correctly. The odd periodic extension has jumps, producing the Gibbs phenomenon in its Fourier sine series near zero and $L$. Every Fourier sine series partial sum is zero at both endpoints, so by approaching $L$ from inside $(0,L)$ one obtainsHence the Fourier sine series approximation cannot converge uniformly even on the whole open interval, despite pointwise convergence at each interior point. The Fourier cosine series approximation has faster coefficient decay and avoids the persistent errors near the endpoints.

The truncated Fourier cosine series gives the better approximation on the closed interval, in the senses of uniform convergence and the asymptotic L2 norm error. To make the comparison precise, let $C_N$ and $S_N$ retain the frequencies $1\le n\le N$, with the constant term also included in $C_N$.

The cosine Fourier coefficients are $O(n^{-2})$. In fact, absolute convergence gives the uniform tail boundFor the Fourier sine series, every Fourier partial sum satisfies $S_N(0)=S_N(\pi )=0$. Thusso there is no uniform convergence to $f$ on the closed interval. Its odd extension has jump discontinuities, producing the Gibbs phenomenon near the endpoints; its Fourier coefficients decrease only like $1/n$.

The advantage persists in the L2 norm, which ignores individual endpoint values. By Parseval's identity and Fourier orthogonality,The nonzero odd cosine coefficients and both parity classes of sine coefficients also give matching lower bounds of these orders by comparison with the corresponding p-series. Hence their L2 norm errors decrease respectively like $N^{-3/2}$ and $N^{-1/2}$. This is a comparison of approximation quality in specified norms, not a claim that one finite sum has smaller error at every individual interior point.

The truncated Fourier cosine series gives the better approximation over the whole closed interval. Its Fourier coefficients obey $|a_n|\le 4/(\pi n^2)$, so absolute convergence and the tail boundgive uniform convergence. The even periodic extension is continuous; its slope changes only cause corners.

By contrast $b_n=(-1{)}^{n+1}/n+O(n^{-2})$. The odd periodic extension has a jump at $\pi$, giving slower coefficient decay and the Gibbs phenomenon near that endpoint. Every truncated Fourier sine series satisfies $S_N(\pi )=0$, soIt therefore cannot converge uniformly to $f$ on $[0,\pi ]$, although it converges at every point of $[0,\pi )$. This comparison concerns approximation across the interval; it does not assert that the cosine truncation has smaller error at every individual interior point for every $N$.

The Fourier cosine series converges to the continuous even $2\pi$-periodic extension. If $r\in [-\pi ,\pi ]$ represents $x$ modulo $2\pi$, its value isIt has value zero at $x=2j\pi$, rises linearly with distance from these points until height $\pi /2$, and then stays flat until the next descending ramp. Thus the sketch on $[-3\pi ,3\pi ]$ has minima at $-2\pi ,0,2\pi$, and plateaux centred at the odd multiples of $\pi$. Its large partial sums round the corners and closely follow the whole graph.

The Fourier sine series converges, away from the odd multiples of $\pi$, to the odd $2\pi$-periodic extensionAt every odd multiple of $\pi$ its value is zero, the midpoint of the jump from $\pi /2$ to $-\pi /2$. At even multiples of $\pi$ it passes continuously through zero. Thus on every interval $((2j-1)\pi ,(2j+1)\pi )$ it has a negative plateau, a rising linear segment through $2j\pi$, and a positive plateau. The jump size is $\pi$.

The blue curves are the partial sums through $n=60$. The open endpoint circles and filled zero dots in the lower panel distinguish the one-sided limits from the value of the Fourier sine series at each jump. Near these jumps the partial sums exhibit the Gibbs phenomenon: the oscillations become concentrated in narrower regions as the number of terms grows, while their limiting overshoot remains about $0.08949$ times the jump, here about $0.281$.

Apply a Fourier transform or discrete Fourier transform, retain the desired low-frequency coefficients and attenuate those in the high-frequency noise band with a low-pass filter, then invert. For a real signal, apply matching gains at positive and negative frequencies to preserve Hermitian symmetry.

Parseval identity for Fourier series shows what an ideal coefficient cutoff does: the output mean-square signal power is the sum of the retained squared coefficients, and cannot exceed that of the unfiltered input. Much of the high-frequency noise can be removed, but a finite cutoff also removes genuine square wave harmonics. Its edges become less sharp, and a sharp spectral cutoff can exhibit the Gibbs phenomenon. There is no claim of exact recovery if the noise overlaps the retained band. Noise already folded into low frequencies by aliasing cannot in general be removed merely by discarding the high-frequency bins afterward.

A square wave alternates between two constant levels for equal halves of a period. Choose a $2\pi$-periodic function equal to $1$ on $(0,\pi )$ and $-1$ on $(-\pi ,0)$, with value zero at the jumps. Its odd function symmetry makes the mean and every cosine Fourier coefficient vanish. Directly integrating the two constant pieces givesThus only odd sine Fourier harmonics occur, with amplitudes decreasing like the reciprocal of their frequency multiplier. Every finite Fourier partial sum is smooth, yet their pointwise limit has jumps. At those jumps the Fourier series converges to zero, the midpoint of the two levels, regardless of which value we assign to the original function there.

This example explains how smooth components can encode sharp transitions. Increasing the number of Fourier harmonics narrows the transition region, but also produces the persistent overshoot described by the Gibbs phenomenon. The illustrated square wave and coefficient calculation are mathematical constructions, not measurements of a physical device.