Generalized Fourier series
In mathematics, in the area of functional analysis, a generalized Fourier series corresponds to the series expansion of square-integrable functions. This series expansion is with respect to an orthogonal basis. A Fourier series uses an orthonormal basis of trigonometric functions. The series expansion is applied to periodic functions. The trigonometric functions are solutions to a Laplacian eigenvalue problem on an interval with periodic boundary conditions. In contrast, a generalized Fourier series can apply to any functions. The functions must satisfy the boundary conditions of a Sturm-Liouville eigenvalue problem on any interval. The series expansion uses any orthogonal basis of solutions to such an eigenvalue problem on that interval with those boundary conditions.
Definition
Consider a set of square-integrable functions values over that are pairwise orthogonal under the inner product
where is a weight function and represents complex conjugation. The generalized Fourier series is then where the coefficients are given by If Φ is an orthogonal basis, then the relation becomes equality in the L2 sense. If Φ is in addition an orthonormal basis, then the denominators for all n.
Examples
Fourier–Legendre series
A function defined on the entire number line is called periodic with period if there is a number such that .
If a function is periodic with period , then it is also periodic with periods , , and so on. Usually, the period of a function is understood as the smallest such number . However, for some functions, arbitrarily small values of exist.
The sequence of functions is known as the trigonometric system. 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.
Let the function be defined on the segment [−π, π]. Given appropriate smoothness and differentiability conditions, 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.
The Legendre polynomials 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. This can be written as a generalized Fourier series (known in this case as a Fourier–Legendre series) involving the Legendre polynomials, so that
As an example, the Fourier–Legendre series may be calculated for over . Then
and a truncated series involving only these terms would be
which differs from by approximately 0.003. 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.
Coefficient theorems
Some theorems on the series coefficients include:
Bessel's inequality
If Φ is a complete basis, then
See also
- Banach space
- Eigenfunctions
- Fractional Fourier transform
- Function space
- Hilbert space
- Least-squares spectral analysis
- Orthogonal function
- Orthogonality
- Topological vector space
- Vector space
References
- Generalized Fourier Series at MathWorld