Английская Википедия:Generalized Fourier series
Шаблон:Use American English Шаблон:Short description Шаблон:Tone In mathematics, a generalized Fourier series is an expansion of a square-integrable function defined on an interval of the real line. The constituent functions of 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 satisfying a Sturm-Liouville eigenvalue problem. These expansions find common use in interpolation theory.[1]
Definition
Consider a set of square-integrable functions with values in <math> \mathbb{F} = \Complex</math> or <math>\mathbb{F} = \R</math>, <math display="block">\Phi = \{\varphi_n:[a,b] \to \mathbb{F}\}_{n=0}^\infty,</math> which are pairwise orthogonal under the inner product <math display="block">\langle f, g\rangle_w = \int_a^b f(x)\,\overline{g}(x)\,w(x)\,dx,</math> where <math>w(x)</math> is a weight function, and <math>\overline\cdot</math> represents complex conjugation, i.e., <math>\overline{g}(x) = g(x)</math> for <math> \mathbb{F} = \R</math>.
The generalized Fourier series of a square-integrable function <math>f : [a, b] \to \mathbb{F}</math>, with respect to Φ, is then <math display="block">f(x) \sim \sum_{n=0}^\infty c_n\varphi_n(x),</math> where the coefficients are given by <math display="block">c_n = {\langle f, \varphi_n \rangle_w\over \|\varphi_n\|_w^2}.</math>
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 <math>\sim </math> becomes equality in the L2 sense, more precisely modulo <math>|\cdot|_w </math> (not necessarily pointwise, nor almost everywhere).
Example (Fourier–Legendre series)
The Legendre polynomials are solutions to the Sturm–Liouville problem
- <math> \left((1-x^2)P_n'(x)\right)'+n(n+1)P_n(x)=0.</math>
As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions with respect to the inner product above with unit weight. So we can form a generalized Fourier series (known as a Fourier–Legendre series) involving the Legendre polynomials, and
- <math>f(x) \sim \sum_{n=0}^\infty c_n P_n(x),</math>
- <math>c_n = {\langle f, P_n \rangle_w\over \|P_n\|_w^2}</math>
As an example, let us calculate the Fourier–Legendre series for <math>f(x)=\cos x</math> over <math>[-1, 1]</math>. Now,
- <math>
\begin{align} c_0 & = {\int_{-1}^1 \cos{x}\,dx \over \int_{-1}^1 (1)^2 \,dx} = \sin{1} \\ c_1 & = {\int_{-1}^1 x \cos{x}\,dx \over \int_{-1}^1 x^2 \, dx} = {0 \over 2/3 } =0 \\ c_2 & = {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \, dx \over \int_{-1}^1 {9x^4-6x^2+1 \over 4} \, dx} = {6 \cos{1} - 4\sin{1} \over 2/5 } \end{align} </math>
and a series involving these terms
- <math>\begin{align}c_2P_2(x)+c_1P_1(x)+c_0P_0(x)&= {5 \over 2} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin1\\
&= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}\end{align}</math>
which differs from <math>\cos x</math> 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
Some theorems on the coefficients cn include:
Bessel's inequality
- <math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2w(x)\,dx.</math>
Parseval's theorem
If Φ is a complete set, then
- <math> \sum_{n=0}^\infty |c_n|^2 = \int_a^b |f(x)|^2w(x)\, dx.</math>
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
