Decompositions of inner product spaces into orthonormal bases
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
Consider a set of square-integrable functions with values in
or
,
![{\displaystyle \Phi =\{\varphi _{n}:[a,b]\to \mathbb {F} \}_{n=0}^{\infty },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f7770ba79954cd5078e4848539c778f84c6afe)
which are pairwise orthogonal under the inner product
![{\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)\,{\overline {g))(x)\,w(x)\,dx,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61b7ef4d07ac3f8a84d444c9ccaf5ef14d769db4)
where
is a weight function, and
represents complex conjugation, i.e.,
for
The generalized Fourier series of a square-integrable function
, with respect to Φ, is then
![{\displaystyle f(x)\sim \sum _{n=0}^{\infty }c_{n}\varphi _{n}(x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21cf3a01f0098d9076982975f35ac549e6cbd39d)
where the coefficients are given by
![{\displaystyle c_{n}={\langle f,\varphi _{n}\rangle _{w} \over \|\varphi _{n}\|_{w}^{2)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19211b8ff8f04cb44de700004a0084d706e3119f)
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)
The Legendre polynomials are solutions to the Sturm–Liouville problem
![{\displaystyle \left((1-x^{2})P_{n}'(x)\right)'+n(n+1)P_{n}(x)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d968e4817e026e2db4fd6e9de8f3f116c5325a99)
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
![{\displaystyle f(x)\sim \sum _{n=0}^{\infty }c_{n}P_{n}(x),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/433050ca54f74ab9c3e55282ffac8b9ef6e0fd41)
![{\displaystyle c_{n}={\langle f,P_{n}\rangle _{w} \over \|P_{n}\|_{w}^{2))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/481cecd0ff19263f284d265cf9fd69fdbced3085)
As an example, we may calculate the Fourier–Legendre series for
over
. We have that,
![{\displaystyle {\begin{aligned}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{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b48ab0a452211dd73275b0a312f8787513e6a2a8)
and a series involving these terms would be
![{\displaystyle {\begin{aligned}c_{2}P_{2}(x)+c_{1}P_{1}(x)+c_{0}P_{0}(x)&={5 \over 2}(6\cos {1}-4\sin {1})\left({3x^{2}-1 \over 2}\right)+\sin 1\\&=\left({45 \over 2}\cos {1}-15\sin {1}\right)x^{2}+6\sin {1}-{15 \over 2}\cos {1}\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33bb6509fc8b7e087be5471f3f789f0109f580)
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
Some theorems on the coefficients
include:
![{\displaystyle \sum _{n=0}^{\infty }|c_{n}|^{2}\leq \int _{a}^{b}|f(x)|^{2}w(x)\,dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b030157028d50ed9125bd15d233411ec6f99faf9)
If Φ is a complete set, then
![{\displaystyle \sum _{n=0}^{\infty }|c_{n}|^{2}=\int _{a}^{b}|f(x)|^{2}w(x)\,dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80ed40fa2ee0ec921e78d0e933061812003c764)