Generalized Fourier series
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In mathematics, a generalized Fourier series expands a square-integrable function defined on an interval of the real line. The constituent functions in the series expansion form an orthonormal basis of an inner product space. While a Fourier series expansion consists only of trigonometric functions, a generalized Fourier series is a decomposition involving any set of functions that satisfy the Sturm-Liouville eigenvalue problem. These expansions find common use in interpolation theory.[1]
Definition[edit]
Consider a set of square-integrable functions with values in or ,
The generalized Fourier series of a square-integrable function , with respect to Φ, is then
If Φ is a complete set, i.e., an orthogonal basis of the space of all square-integrable functions on [a, b], as opposed to a smaller orthogonal set, the relation becomes equality in the L2 sense, more precisely modulo (not necessarily pointwise, nor almost everywhere).
Example (Fourier–Legendre series)[edit]
The Legendre polynomials are solutions to the Sturm–Liouville problem
As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions with respect to the inner product above with unit weight. We can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and
As an example, we may calculate the Fourier–Legendre series for over . We have that,
and a series involving these terms would be
which differ from by approximately 0.003. It may be advantageous to use such Fourier–Legendre series since the eigenfunctions are all polynomials and hence the integrals and thus the coefficients are easier to calculate.
Coefficient theorems[edit]
Some theorems on the coefficients include:
Bessel's inequality[edit]
Parseval's theorem[edit]
If Φ is a complete set, then
See also[edit]
- Banach space
- Eigenfunctions
- Fractional Fourier transform
- Function space
- Hilbert space
- Least-squares spectral analysis
- Orthogonal function
- Orthogonality
- Topological vector space
- Vector space
References[edit]
- ^ Howell, Kenneth B. (2001-05-18). Principles of Fourier Analysis. Boca Raton: CRC Press. doi:10.1201/9781420036909. ISBN 978-0-429-12941-4.