mathworld.wolfram.com

Generalized Fourier Series -- from Wolfram MathWorld

️Weisstein, Eric W.
️Sat Jan 03 2004

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld

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 , the functions satisfy an orthogonality relationship of the form

int_Rphi_m(x)phi_n(x)w(x)dx=c_mdelta_(mn)

(1)

over a range , 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

(2)

and plug this into the orthogonality relationships to obtain

int_Rf(x)phi_n(x)w(x)dx
=int_Rsum_(n=0)^inftya_nphi_m(x)phi_n(x)w(x)dx
=sum_(n=0)^inftya_nintphi_m(x)phi_n(x)w(x)dx
=sum_(n=0)^inftya_nc_mdelta_(mn)
=a_nc_n.

(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 f(x) 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 f_1(n,x) and f_2(n,x) satisfy orthogonality relationships of the form

for m,n!=0 over a range , 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

f(x)=sum_(n=0)^inftya_nf_1(n,x)+sum_(n=0)^inftyb_nf_2(n,x)
=f_1(0)a_0+sum_(n=1)^inftya_nf_1(n,x)+f_2(0)b_0+sum_(n=1)^inftyb_nf_2(n,x)
=[f_1(0)a_0+f_2(0)b_0]+sum_(n=1)^inftya_nf_1(n,x)+sum_(n=1)^inftyb_nf_2(n,x)
=e+sum_(n=1)^inftya_nf_1(n,x)+sum_(n=1)^inftyb_nf_2(n,x)

(10)

and plug this into the orthogonality relationships to obtain

int_Rf(x)f_1(n,x)w(x)dx=eint_Rf_1(n,x)dx+int_Rsum_(m=1)^inftya_mf_1(m,x)f_1(n,x)w(x)dx+int_Rsum_(m=1)^inftyb_mf_1(m,x)f_2(n,x)w(x)dx
=e·0+sum_(m=1)^inftya_mint_Rf_1(m,x)f_1(n,x)w(x)dx+sum_(m=1)^inftyb_mint_Rf_1(m,x)f_2(n,x)w(x)dx
=sum_(m=1)^inftya_mc_mdelta_(mn)+sum_(m=1)^inftyb_m·0
=a_nc_n
int_Rf(x)f_2(n,x)w(x)dx=eint_Rf_2(n,x)dx+int_Rsum_(m=1)^inftya_mf_1(m,x)f_2(n,x)w(x)dx+int_Rsum_(m=1)^inftyb_mf_2(m,x)f_2(n,x)w(x)dx
=e·0+sum_(m=1)^inftya_mint_Rf_1(m,x)f_2(n,x)w(x)dx+sum_(m=1)^inftyb_mint_Rf_2(m,x)f_2(n,x)w(x)dx
=sum_(m=1)^inftya_m·0+sum_(m=1)^inftyb_md_mdelta_(mn)
=b_nd_n
int_Rf(x)w(x)dx=eint_Rdx+int_Rsum_(m=1)^inftya_mf_1(m,x)w(x)dx+int_Rsum_(m=1)^inftyb_mf_2(m,x)w(x)dx
=eint_Rdx+sum_(m=1)^inftya_mint_Rf_1(m,x)w(x)dx+sum_(m=1)^inftyb_nint_Rf_2(m,x)w(x)dx
=eint_Rdx+sum_(m=1)^inftya_m·0+sum_(m=1)^inftyb_m·0
=eint_Rdx.

(11)

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,