Generalized Fourier series

From Academic Kids

In mathematical analysis, there are many potentially useful generalizations of Fourier series. For a set of square-integrable, pairwise-orthogonal (with respect to some weight function w(x)) functions

<math>\Phi = \{\varphi_n:[a,b]\rightarrow F\}_{n=0}^\infty,<math>

the generalized Fourier series of a square-integrable function f:[a, b] → F is

<math>f(x) \sim \sum_{n=0}^\infty c_n\varphi_n(x),<math>

where the coefficients are given by

<math>c_n = {\langle f, \varphi_n \rangle_w\over ||\varphi_n||_w^2}<math>

where the inner product is the conventional one for functions. Where F = C, this is

<math>\langle f, g\rangle_w = \int_a^b f(x)\overline{g}(x)w(x)\,dx<math>

where <math>\overline{g}(x)<math> represents the complex conjugate of <math>g(x)\,\!<math>. If F = R, the complex conjugate is real, so

<math>\langle f, g\rangle_w = \int_a^b f(x)g(x)w(x)\,dx<math>

The relation <math>\sim<math> becomes equality if Φ is a complete set, i.e., an orthonormal basis of the space of all square-integrable functions on [a, b], as opposed to a smaller orthonormal set, provided the convergence of the series is understood to be convergence in mean square and not necessarily pointwise convergence, nor convergence almost everywhere.

Contents

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>

and because of the theory, these polynomials are eigenfunctions of the problem and are solutions are orthogonal 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\varphi_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 f(x)=cos x over [−1,1]. Now,

<math>c_0 = \sin{1} = {\int_{-1}^1 \cos{x} \over \int_{-1}^1 (1)^2}<math>
<math>c_1 = 0 = {\int_{-1}^1 x \cos{x} \over \int_{-1}^1 x^2} = {0 \over 2/3 }<math>
<math>c_2 = {5 \over 6} (6 \cos{1} - 4\sin{1}) = {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \over \int_{-1}^1 {9x^4-6x^2+1 \over 4}} = {6 \cos{1} - 4\sin{1} \over 2/5 }<math>

and a series involving these terms

<math>c_2P_2(x)+c_1P_1(x)+c_0P_0(x)= {5 \over 6} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin{1}(1)<math>
<math>= ({45 \over 2} \cos{1} - 15 \sin{1})x^2+6 \sin{1} - {15 \over 2}\cos{1}<math>

which differs from cos x by approximately 0.003, about 0. 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)|^2\,dx.<math>

Parseval's theorem

If Φ is a complete set,

<math>\sum_{n=0}^\infty |c_n|^2 = \int_a^b|f(x)|^2\, dx.<math>

See also

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