Generalized Fourier Series

A generalized Fourier series is a series expansion in a complete orthogonal system, with convergence in the relevant L²-norm. The prototypical example is the Fourier series based on the constant function 1 and the functions and , , on . Another example is the Laplace series, a double series expansion in the spherical harmonics . Their inner product uses complex conjugation and the area element over and (NIST, §14.30(ii)).

For a complete orthogonal system of nonzero univariate functions on an interval , take a nonnegative measurable function as weight. Work in the L²-space , identifying functions equal almost everywhere with respect to . The orthogonality relationship is

(1)

where and is the Kronecker delta. For , the expansion

(2)

means that the partial sums converge to in norm (Axler 2026, pp. 242-244). For fixed and , finite summation gives

(3)

Passing to the limit is justified by Schwarz's inequality, since . Thus the expansion coefficients are

(4)

This argument uses finite sums before a norm limit, and does not require interchanging an infinite series and an integral.

A useful special form occurs when the constant function 1 and two families and together form a complete orthogonal system in . Assume , and write the orthogonality relationships as

			(5)
			(6)
			(7)
			(8)
			(9)

for , where . Define the finite partial sum

(10)

For , the corresponding calculations are

(11)

Taking limits in the inner product, as above, gives

			(12)
			(13)
			(14)

The usual Fourier series is recovered by taking and for , together with the constant function 1, over with weight . For every positive integer ,

			(15)
			(16)

Therefore, the Fourier series of a function is given by

(17)

where the coefficients are

			(18)
			(19)
			(20)

A complete biorthogonal system generalizes the coefficient pairing beyond orthogonal functions. Its completeness alone does not assert that the associated series converges in norm to every vector; a Schauder basis requires that additional expansion property (Heil 1998, pp. 50-51).