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Complete Orthogonal System

A set of orthogonal 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=||f-(c_1phi_1+...+c_nphi_n)||^2

(where ||f|| denotes the L2-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->infty)int_a^b[f(x)-sum_(n=0)^ma_nphi_n(x)]^2w(x)dx=0,

where the above integral is a Lebesgue integral.

Examples of complete orthogonal systems include {sin(nx),cos(nx)} over [-pi,pi] (which actually form a slightly more special type of system known as a complete biorthogonal system), the Legendre polynomials {P_n(x)} over [-1,1] (Kaplan 1992, p. 512), and {sqrt(x)J_0(alpha_nx)} on [0,1], where J_0(z) is a Bessel function of the first kind and alpha_n is its nth root (Kaplan 1992, p. 514). These systems lead to the Fourier series, Fourier-Legendre series, and Fourier-Bessel series, respectively.

See also

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

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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.

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Complete Orthogonal System

Cite this as:

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

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