Generalized Fourier Series

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)

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)

and plug this into the orthogonality relationships to obtain

(3)

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)

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)
			(6)
			(7)
			(8)
			(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)
			(13)
			(14)

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)
			(16)

Therefore, the Fourier series of a function is given by

(17)

where the coefficients are

			(18)
			(19)
			(20)