Fourier Series · aiwiki.page

A Fourier series is an infinite series that expresses a periodic function in terms of sines and cosines, or equivalently complex exponentials. Each component has a frequency that is an integer multiple of a fundamental frequency, with coefficients specifying its contribution. Fourier series are central to harmonic analysis and provide tools for studying oscillations, approximating functions, and solving equations. Their interpretation requires distinguishing different kinds of convergence: a series may reproduce a function in mean square without converging to its assigned value at every point. (ocw.mit.edu)

Definition and coefficients

For an integrable, real-valued function $f$ with period $2\pi$ , the trigonometric Fourier series is written

f(x)\sim \frac{a_0}{2} +\sum_{n=1}^{\infty} \bigl[a_n\cos(nx)+b_n\sin(nx)\bigr],

where

a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)\,dx \quad(n\geq0),

b_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\sin(nx)\,dx \quad(n\geq1).

The symbol $\sim$ indicates the associated expansion without asserting convergence everywhere. These integrals measure the contribution of each sinusoidal component. The constant $a_0/2$ is the average value over one period. For period $T$ , the corresponding arguments are $2\pi nx/T$ , and the coefficient normalization becomes $2/T$ . (ocw.mit.edu)

Using complex numbers and the exponential function, the same expansion takes the form

f(x)\sim\sum_{n\in\mathbb Z}c_ne^{inx}, \qquad c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}\,dx.

Here $c_0=a_0/2$ , while $c_n=(a_n-ib_n)/2$ for positive $n$ . For real-valued functions, $c_{-n}=\overline{c_n}$ . Negative indices encode the conjugate components needed to produce a real-valued result. (ocw.mit.edu)

Orthogonality and geometric interpretation

The coefficient formulas follow from orthogonality: distinct sine and cosine modes have zero integrated product over a full period. Thus multiplying an expansion by one mode and integrating isolates its coefficient. Even functions have no sine terms; odd functions have neither cosine terms nor a constant term, when the interval is centered at zero. (ocw.mit.edu)

This construction has a geometric interpretation in the Hilbert space $L^2([-\pi,\pi])$ , an instance of an Lp space. With the normalized inner product

\langle f,g\rangle= \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)\overline{g(x)}\,dx,

the functions $e^{inx}$ form a complete orthonormal basis. The partial sum

S_Nf=\sum_{n=-N}^{N}c_ne^{inx}

is the orthogonal projection of $f$ onto their finite-dimensional span. It minimizes integrated squared error among all trigonometric polynomials containing those modes. (math.mit.edu)

Parseval’s identity states that

\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^2\,dx =\sum_{n\in\mathbb Z}|c_n|^2.

It identifies the squared norm of a function with the sum of the squared magnitudes of its Fourier coefficients, extending the geometry of orthogonal coordinates to functions. (math.mit.edu)

Convergence and discontinuities

Convergence is not a single property. For every $f\in L^2$ , Fourier partial sums converge in mean square:

\int_{-\pi}^{\pi}|f(x)-S_Nf(x)|^2\,dx\longrightarrow0.

This does not by itself guarantee pointwise convergence at every point or uniform convergence. Under standard sufficient conditions, such as piecewise continuous differentiability, the series converges at each point to

\frac{f(x^-)+f(x^+)}{2}.

It therefore recovers the function at continuity points but approaches the midpoint of a jump. Endpoint behavior is determined by the periodic extension, so unequal endpoint limits create a jump even if the function is smooth inside the interval. (ocw.mit.edu)

Near a jump, partial sums exhibit the Gibbs phenomenon: oscillations and overshoot persist in magnitude as more terms are included, although the affected region narrows. Averaging the partial sums changes this behavior. Fejér’s theorem guarantees that their arithmetic means converge uniformly for every continuous periodic function, even though continuity alone does not guarantee uniform convergence of ordinary Fourier partial sums. (ocw.mit.edu)

Example and approximation properties

Consider the periodic square wave defined by $f(x)=1$ for $0<x<\pi$ and $f(x)=-1$ for $-\pi<x<0$ . Substitution into the coefficient formulas gives

f(x)\sim\frac{4}{\pi} \left(\sin x+\frac{\sin3x}{3} +\frac{\sin5x}{5}+\cdots\right).

Only odd-numbered sine modes occur. Away from the jumps the series approaches $1$ or $-1$ ; at the jumps it approaches zero, the midpoint of the one-sided limits. Finite approximations display Gibbs oscillations. (ocw.mit.edu)

Smoothness controls coefficient decay and hence approximation quality. Repeated integration by parts relates high-frequency coefficients to derivatives, provided the periodic boundary terms match appropriately. Smooth periodic functions have rapidly decaying coefficients, while jumps generally cause slower decay. This relationship underlies highly accurate Fourier approximations in numerical analysis. (ocw.mit.edu)

Origins, applications, and related transforms

Joseph Fourier developed trigonometric expansions in his study of heat conduction, publishing his principal results in Théorie analytique de la chaleur in 1822. This work connected the representation of functions with physical evolution governed by equations. (gresham.ac.uk)

For the heat equation on a periodic domain, expansion into Fourier modes converts a partial differential equation into separate equations for the coefficients. Higher-frequency modes decay faster, explaining the smoothing of an initial temperature distribution. Related expansions support other boundary-value problems. In signal processing, Fourier coefficients describe periodic signals through their frequency components. (see.stanford.edu)

A Fourier transform generally describes nonperiodic functions through a continuous frequency variable, whereas a Fourier series uses discrete harmonic frequencies. For finitely sampled data, the discrete Fourier transform supplies a finite counterpart. The fast Fourier transform is an efficient algorithm for computing that finite transform, rather than a different mathematical transform. (ocw.mit.edu)