TOPICS

# Complete Orthogonal System

A set of orthogonal functions is termed complete in the closed interval if, for every piecewise continuous function in the interval, the minimum square error

(where denotes the L2-norm with respect to a weighting function ) converges to zero as becomes infinite. Symbolically, a set of functions is complete if

where the above integral is a Lebesgue integral.

Examples of complete orthogonal systems include over (which actually form a slightly more special type of system known as a complete biorthogonal system), the Legendre polynomials over (Kaplan 1992, p. 512), and on , where is a Bessel function of the first kind and is its th root (Kaplan 1992, p. 514). These systems lead to the Fourier series, Fourier-Legendre series, and Fourier-Bessel series, respectively.

Bessel's Inequality, Complete Biorthogonal System, Complete Set of Functions, Fourier Series, Generalized Fourier Series, Hilbert Space, L2-Norm, Orthogonal Functions, Orthonormal Functions, Overcomplete System, Parseval's Theorem

## Explore with Wolfram|Alpha

More things to try:

## References

Arfken, G. "Completeness of Eigenfunctions." §9.4 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 523-538, 1985.Kaplan, W. "Fourier Series of Orthogonal Functions: Completeness" and "Sufficient Conditions for Completeness." §7.11 and 7.12 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 501-505, 1992.

## Referenced on Wolfram|Alpha

Complete Orthogonal System

## Cite this as:

Weisstein, Eric W. "Complete Orthogonal System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CompleteOrthogonalSystem.html