complete functions

A set of orthogonal functions {ϕ_n(x)} is termed complete in the closed interval x element [a, b] if, for every piecewise continuous function f(x) in the interval, the minimum square error E_n congruent ( left double bracketing bar f - (c_1 ϕ_1 + ... + c_n ϕ_n) right double bracketing bar )^2 (where left double bracketing bar f right double bracketing bar denotes the L^2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite. Symbolically, a set of functions is complete if lim_(m->∞) integral_a^b [f(x) - sum_(n = 0)^m a_n ϕ_n(x)]^2 w(x) d x = 0, where the above integral is a Lebesgue integral.

Bessel's inequality | complete biorthogonal system | Fourier series | generalized Fourier series | Hilbert space | L^2-norm | orthogonal functions | orthonormal functions | Parseval's theorem