A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions.
The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion is applied to periodic functions.
In contrast, a generalized Fourier series uses any set of orthogonal basis functions and can apply to any square integrable function.
of square-integrable complex valued functions defined on the closed interval
that are pairwise orthogonal under the weighted inner product:
Then, the generalized Fourier series of a function
of square integrable functions defined on a given interval, one can find orthogonal bases by considering a class of boundary value problems on the interval
called regular Sturm-Liouville problems.
are self-adjoint boundary conditions, and
is a positive continuous functions on
Given a regular Sturm-Liouville problem as defined above, the set
of eigenfunctions corresponding to the distinct eigenvalue solutions to the problem form an orthogonal basis for
that satisfies the boundary conditions of this Sturm-Liouville problem, the series
exists such that, for any real number
Usually, the period of a function is understood as the smallest such number
However, for some functions, arbitrarily small values of
Any linear combination of functions of a trigonometric system, including an infinite combination (that is, a converging infinite series), is a periodic function with a period of 2π.
On any segment of length 2π (such as the segments [−π,π] and [0,2π]) the trigonometric system is an orthogonal system.
This means that for any two functions of the trigonometric system, the integral of their product over a segment of length 2π is equal to zero.
This integral can be treated as a scalar product in the space of functions that are integrable on a given segment of length 2π.
be defined on the segment [−π, π].
may be represented on this segment as a linear combination of functions of the trigonometric system, also referred to as the expansion of the function
into a trigonometric Fourier series.
are solutions to the Sturm–Liouville eigenvalue problem As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions with respect to the inner product with unit weight.
Then and a truncated series involving only these terms would be which differs from
In computational applications it may be advantageous to use such Fourier–Legendre series rather than Fourier series since the basis functions for the series expansion are all polynomials and hence the integrals and thus the coefficients may be easier to calculate.
Some theorems on the series' coefficients
include: Bessel's inequality is a statement about the coefficients of an element
in a Hilbert space with respect to an orthonormal sequence.
Bessel in 1828:[5] Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.