A generalized Fourier series is a series expansion of a function based on the special properties of a complete orthogonal system
of functions. The prototypical example of such a series is the Fourier
series, which is based of the biorthogonality of the functions and
(which form a complete
biorthogonal system under integration over the range
). Another common example is the Laplace
series, which is a double series expansion based on the orthogonality of the
spherical harmonics
over
and
.
Given a complete orthogonal system of univariate functions over the interval
, the functions
satisfy an orthogonality relationship of the form
(1)
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over a range ,
where
is a weighting function,
are given constants and
is the Kronecker delta.
Now consider an arbitrary function
. Write it as a series
(2)
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and plug this into the orthogonality relationships to obtain
(3)
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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
(4)
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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
(5)
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(6)
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(7)
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(8)
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(9)
|
for
over a range
,
where
and
are given constants and
is the Kronecker delta.
Now consider an arbitrary function
and write it as a series
(10)
|
and plug this into the orthogonality relationships to obtain
(11)
|
As a result of these relations, if a series for of the assumed form exists, its coefficients will satisfy
(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
with weighting function
and noting that, for this choice of functions,
(15)
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(16)
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Therefore, the Fourier series of a function is given by
(17)
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where the coefficients are
(18)
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(19)
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(20)
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