Generalized Fourier series

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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

functions with values in ${\displaystyle \mathbb {F} =\mathbb {C} }$ or ${\displaystyle \mathbb {F} =\mathbb {R} }$, which are pairwise where ${\displaystyle w(x)}$ is a weight function, and ${\displaystyle {\overline {\cdot }}}$ represents , i.e., ${\displaystyle {\overline {g}}(x)=g(x)}$ for ${\displaystyle \mathbb {F} =\mathbb {R} }$.

The generalized Fourier series of a

function ${\displaystyle f:[a,b]\to \mathbb {F} }$, with respect to Φ, is then where the coefficients are given by

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

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, calculating the Fourier–Legendre series for ${\displaystyle f(x)=\cos x}$ over ${\displaystyle [-1,1]}$. Now,

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

and a series involving these terms

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

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

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