http://mathworld.wolfram.com/GeneralizedFourierSeries.html
A generalized Fourier series is a series expansion of a function based on the special properties of acomplete orthogonal system of functions. The prototypical example of such a series is theFourier series, which is based of the biorthogonality of the functions and
(which form acomplete biorthogonal system under integration over the range
. Another common example is theLaplace series, which is a double series expansion based on the orthogonality of thespherical harmonics
over
and
.
Given a complete orthogonal system of univariate functions over the interval
, the functions
satisfy an orthogonality relationship of the form
over a range , where
is aweighting function,
are given constants and
is theKronecker delta. Now consider an arbitrary function
. Write it as a series
and plug this into the orthogonality relationships to obtain
Note that the order of integration and summation has been reversed in deriving the above equations. As a result of these relations, if a series for of the assumed form exists, its coefficients will satisfy
Given a complete biorthogonal system of univariate functions, the generalized Fourier series takes on a slightly more special form. In particular, for such a system, the functions and
satisfy orthogonality relationships of the form
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(5)
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(7)
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(8)
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(9)
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for over a range
, where
and
are given constants and
is theKronecker delta. Now consider an arbitrary function
and write it as a series
and plug this into the orthogonality relationships to obtain
As a result of these relations, if a series for of the assumed form exists, its coefficients will satisfy
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(12)
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(13)
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(14)
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The usual Fourier series is recovered by taking and
which form a complete orthogonal system over
withweighting function
and noting that, for this choice of functions,
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(15)
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(16)
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Therefore, the Fourier series of a function is given by
where the coefficients are
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(19)
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(20)
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