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the generalized Fourier series of a square-integrable function f:[a, b] → F is
where the coefficients are given by
where the inner product is the conventional one for functions. Where F = C, this is
where represents the complex conjugate of . If F = R, the complex conjugate is real, so
The relation 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.
The Legendre polynomials are solutions to the Sturm-Liouville problem
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
As an example, let us calculate the Fourier-Legendre series for f(x)=cos x over [−1,1]. Now,
and a series involving these terms
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 cn include:
If Φ is a complete set,