Fourier-Legendre Series

A Fourier-Legendre series is an expansion in the Legendre polynomials, which form a complete orthogonal system on with weighting function . Every function has the expansion

(1)

with convergence in the norm . Functions equal almost everywhere are identified; this statement does not require pointwise convergence at every (National Institute of Standards and Technology 2026, §18.2(viii)).

To obtain the coefficients, first use the finite partial sum . Multiplication by and integration give

(2)

The Legendre polynomials obey the orthogonality relationship

(3)

where is the Kronecker delta (National Institute of Standards and Technology 2026, Table 18.3.1). For , this gives

			(4)
			(5)

The inner product with is continuous in this norm by Schwarz's inequality. Passing to the limit therefore yields

(6)

For example, for , the first few terms of the Fourier-Legendre series are

(7)