Generalized Fourier series

Consider a set of square-integrable functions with values in ${\displaystyle \mathbb {F} =\mathbb {C} {\mbox{ or }}\mathbb {R} }$,

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

which are pairwise orthogonal for 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 f: [a, b] → ${\displaystyle \mathbb {F} }$, 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).

The Legendre polynomials are solutions to the Sturm–Liouville problem

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

and because of Sturm-Liouville theory, these polynomials are eigenfunctions of the problem and are solutions 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

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

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

As an example, let us calculate the Fourier–Legendre series for ƒ(x) = cos x over [−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

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

${\displaystyle =\left({45 \over 2}\cos {1}-15\sin {1}\right)x^{2}+6\sin {1}-{15 \over 2}\cos {1}}$

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.

Some theorems on the coefficients c_n include:

• Fractional Fourier transform
• Orthogonality
• Orthogonal function
• Eigenfunctions
• Vector space
• Function space
• Topological vector space
• Hilbert space
• Banach space