The generalized Fourier series of a square-integrable function f: →, with respect to Φ, is then
where the coefficients are given by
If Φ is a complete set, i.e., an orthonormal basis of the space of all square-integrable functions on, as opposed to a smaller orthonormal set, the relation becomes equality in the L² sense, more precisely modulo |·|w (not necessarily pointwise, nor almost everywhere).
Read more about Generalized Fourier Series: Example (Fourier–Legendre Series)
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