Generalized Fourier series

In mathematics, a generalized Fourier series is a method of expanding 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 ${\displaystyle \mathbb {F} =\mathbb {C} }$ or ${\displaystyle \mathbb {F} =\mathbb {R} }$,

$${\displaystyle \Phi =\{\varphi _{n}:[a,b]\to \mathbb {F} \}_{n=0}^{\infty },}$$

which are pairwise orthogonal under the inner product

$${\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)\,{\overline {g}}(x)\,w(x)\,dx,}$$

where ${\displaystyle w(x)}$ is a weight function, and ${\displaystyle {\overline {\cdot }}}$ represents complex conjugation, i.e., ${\displaystyle {\overline {g}}(x)=g(x)}$ for ${\displaystyle \mathbb {F} =\mathbb {R} }$.

The generalized Fourier series of a square-integrable function ${\displaystyle f:[a,b]\to \mathbb {F} }$, with respect to Φ, is then

$${\displaystyle f(x)\sim \sum _{n=0}^{\infty }c_{n}\varphi _{n}(x),}$$

where the coefficients are given by

$${\displaystyle c_{n}={\langle f,\varphi _{n}\rangle _{w} \over \|\varphi _{n}\|_{w}^{2}}.}$$

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 ${\displaystyle \sim }$ becomes equality in the L² sense, more precisely modulo ${\displaystyle |\cdot |_{w}}$ (not necessarily pointwise, nor almost everywhere).

Example (Fourier–Legendre series)

The Legendre polynomials are solutions to the Sturm–Liouville problem

${\displaystyle \left((1-x^{2})P_{n}'(x)\right)'+n(n+1)P_{n}(x)=0.}$

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

${\displaystyle f(x)\sim \sum _{n=0}^{\infty }c_{n}P_{n}(x),}$

${\displaystyle c_{n}={\langle f,P_{n}\rangle _{w} \over \|P_{n}\|_{w}^{2}}}$

As an example, we may calculate the Fourier–Legendre series for ${\displaystyle f(x)=\cos x}$ over ${\displaystyle [-1,1]}$. We have that,

${\displaystyle {\begin{aligned}c_{0}&={\int _{-1}^{1}\cos {x}\,dx \over \int _{-1}^{1}(1)^{2}\,dx}=\sin {1}\\c_{1}&={\int _{-1}^{1}x\cos {x}\,dx \over \int _{-1}^{1}x^{2}\,dx}={0 \over 2/3}=0\\c_{2}&={\int _{-1}^{1}{3x^{2}-1 \over 2}\cos {x}\,dx \over \int _{-1}^{1}{9x^{4}-6x^{2}+1 \over 4}\,dx}={6\cos {1}-4\sin {1} \over 2/5}\end{aligned}}}$

and a series involving these terms would be

${\displaystyle {\begin{aligned}c_{2}P_{2}(x)+c_{1}P_{1}(x)+c_{0}P_{0}(x)&={5 \over 2}(6\cos {1}-4\sin {1})\left({3x^{2}-1 \over 2}\right)+\sin 1\\&=\left({45 \over 2}\cos {1}-15\sin {1}\right)x^{2}+6\sin {1}-{15 \over 2}\cos {1}\end{aligned}}}$

which differs from ${\displaystyle \cos x}$ 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 ${\displaystyle c_{n}}$ include:

Bessel's inequality

${\displaystyle \sum _{n=0}^{\infty }|c_{n}|^{2}\leq \int _{a}^{b}|f(x)|^{2}w(x)\,dx.}$

Parseval's theorem

If Φ is a complete set, then

${\displaystyle \sum _{n=0}^{\infty }|c_{n}|^{2}=\int _{a}^{b}|f(x)|^{2}w(x)\,dx.}$

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

1. Howell, Kenneth B. (2001-05-18). Principles of Fourier Analysis. Boca Raton: CRC Press. doi:10.1201/9781420036909. ISBN   978-0-429-12941-4.

• https://mathworld.wolfram.com/GeneralizedFourierSeries.html