Fourier expansion

Many other Fourier-related transforms have since been defined.

Definition of Fourier series

Suppose f(x) is a complex-valued function of a real number, is periodic with period 2π, and is square-integrable over the interval from 0 to 2π. Let

Then the Fourier series representation of f(x) is given by

Since

this is equivalent to representing f(x) as a infinite linear combination of functions of the form cos(nx) and sin(nx), i.e.

Convergence of Fourier series

While the coefficients a_n and b_n can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f.

A partial answer is that if f is square-integrable then

(this is convergence in the norm of the space $L^2$).

That much was proved in the 19th century, as was the fact that if f is piecewise continuous then the series converges at each point of continuity. Perhaps surprisingly, it was not shown until the 1960s that if f is quadratically integrable then the series converges for every value of x except those in some set of measure zero.

General formulation

The useful properties of Fourier series are largely derived from the orthogonality of the functions e^{i n x}. Other sequences of orthogonal functions have similar properties. Examples include sequences of Bessel functions and orthogonal polynomials Such sequences are commonly the solutions of a differential equation; a large class of useful sequences are solutions of the so-called Sturm-Liouville problems.

See also

• orthogonal series
• Fourier transform
• harmonic analysis
• Gibbs phenomenon

Referenced By