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

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

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

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

As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions 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, 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.

Some theorems on the coefficients c_n include:

Bessel's inequality edit

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

Parseval's theorem edit

If Φ is a complete set, then

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

• Banach space
• Eigenfunctions
• Fractional Fourier transform
• Function space
• Hilbert space
• Least-squares spectral analysis
• Orthogonal function
• Orthogonality
• Topological vector space
• Vector space

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