Complete Functions

A set of Orthonormal Functions ${\phi_n(x)}$ is termed complete in the Closed Interval $x\in[a,b]$ if, for every piecewise Continuous Function $f(x)$ in the interval, the minimum square error

$$E_n \equiv \Vert f-(c_1\phi_1+\ldots+c_n\phi_n)\Vert^2$$

(where $\Vert f\Vert$ denotes the Norm) converges to zero as $n$ becomes infinite. Symbolically, a set of functions is complete if

$$\lim_{m\to \infty} \int_a^b \left[{f(x) - \sum_{n=0}^ma_n\phi_n(x)}\right]^2 w(x)\,dx = 0,$$

where $w(x)$ is a Weighting Function and the above is a Lebesgue Integral.

See also Bessel's Inequality, Hilbert Space

References

Arfken, G. ``Completeness of Eigenfunctions.'' §9.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 523-538, 1985.