Generalized Fourier series

In mathematical analysis, there are many potentially useful generalizations of Fourier series. For a set of square-integrable, pairwise-orthogonal (with respect to some weight function w(x)) functions

Related Topics:
Mathematical analysis - Fourier series - Square-integrable - Pairwise-orthogonal - Weight function

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:Phi = {arphi_n: ightarrow F}_{n=0}^infty,

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the generalized Fourier series of a square-integrable function f: → F is

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:f(x) sim sum_{n=0}^infty c_narphi_n(x),

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where the coefficients are given by

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:c_n = {langle f, arphi_n angle_wover ||arphi_n||_w^2}

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where the inner product is the conventional one for functions. Where F = C, this is

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:langle f, g angle_w = int_a^b f(x)overline{g}(x)w(x),dx

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where overline{g}(x) represents the complex conjugate of g(x),!. If F = R, the complex conjugate is real, so

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:langle f, g angle_w = int_a^b f(x)g(x)w(x),dx

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The relation sim becomes equality 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, provided the convergence of the series is understood to be convergence in mean square and not necessarily pointwise convergence, nor convergence almost everywhere.

Related Topics:
Orthonormal basis - Almost everywhere

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