In mathematics, a Fourier series, named in honor of Joseph Fourier (1768-1830), is a representation of a periodic function as a sum of periodic functions of the form
Table of contents |
2 Convergence of Fourier series 3 General formulation 4 See also |
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
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
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.
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.