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Complete Orthogonal System

A complete orthogonal system in a Hilbert space H is a family of nonzero, pairwise orthogonal vectors whose vector space span is a dense set in H (Axler 2026, pp. 242-244).

For functions on a closed interval [a,b], a<b, take the L2-space H=L^2([a,b],w(x)dx) with a nonnegative measurable weight w. Here functions equal almost everywhere with respect to w(x)dx are identified, and the inner product is <f,g>=int_a^bf(x)g(x)^_w(x)dx. For a sequence {phi_n}_(n=0)^infty of nonzero orthogonal functions in H, the coefficients

a_n=(<f,phi_n>)/(||phi_n||^2)
(1)

give the minimum square error

E_m=||f-sum_(n=0)^ma_nphi_n||^2=int_a^b|f(x)-sum_(n=0)^ma_nphi_n(x)|^2w(x)dx.
(2)

The norm is the L2-norm, and the system is complete iff

lim_(m->infty)E_m=0
(3)

for every f in H. This is convergence in norm, rather than a requirement of pointwise convergence.

Examples with weight 1 include {1,sin(nx),cos(nx):n>=1} on [-pi,pi], the Legendre polynomials {P_n(x)}_(n=0)^infty on [-1,1] (Kaplan 1992, p. 512), and {sqrt(x)J_0(alpha_nx)}_(n=1)^infty on [0,1], where alpha_n is the nth positive root of the Bessel function of the first kind J_0 (Kaplan 1992, p. 514; NIST 2026). 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.Axler, S. Measure, Integration & Real Analysis. Author-hosted revision, June 12, 2026. https://measure.axler.net/MIRA.pdf.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.National Institute of Standards and Technology. "General Orthogonal Polynomials," "Integrals," and "Sums." §§18.2(viii), 10.22(ii), and 10.23(iii) in Digital Library of Mathematical Functions. Version 1.2.8, 2026. https://dlmf.nist.gov/18.2.viii. https://dlmf.nist.gov/10.22.E37. https://dlmf.nist.gov/10.23.iii.

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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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