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 cos(nx) and sin(nx) (which form a complete biorthogonal system under integration over the range [-pi,pi]). Another common example is the Laplace series, which is a double series expansion based on the orthogonality of the spherical harmonics Y_l^m(theta,phi) over theta in [0,pi] and phi in [0,2pi].

Given a complete orthogonal system of univariate functions {phi_n(x)} over the interval R, the functions phi_n(x) satisfy an orthogonality relationship of the form

over a range R, where w(x) is a weighting function, c_m are given constants and delta_(mn) is the Kronecker delta. Now consider an arbitrary function f(x). 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 f(x) 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 f_1(n,x) and f_2(n,x) satisfy orthogonality relationships of the form

for m,n!=0 over a range R, where c_m and d_m are given constants and delta_(mn) is the Kronecker delta. Now consider an arbitrary function f(x) 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 f(x) of the assumed form exists, its coefficients will satisfy

The usual Fourier series is recovered by taking f_1(n,x)=cos(nx) and f_2(n,x)=sin(nx) which form a complete orthogonal system over [-pi,pi] with weighting function w(x)=1 and noting that, for this choice of functions,

Therefore, the Fourier series of a function f(x) is given by

where the coefficients are

See also

Bessel Function Neumann Series, Bessel's Inequality, Fourier-Bessel Series, Fourier-Legendre Series, Fourier Series, Generalized Fourier Integral, Kapteyn Series, Orthogonal Basis, Orthogonal Polynomials, Orthonormal Basis, Laplace Series, Parseval's Theorem

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Cite this as:

Weisstein, Eric W. "Generalized Fourier Series." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedFourierSeries.html

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