Bessel's Inequality

If f(x) is piecewise continuous and has a generalized Fourier series

with weighting function w(x), it must be true that

But the coefficient of the generalized Fourier series is given by

so

Equation (6) is an inequality if the functions {phi_i} do not form a complete orthogonal system. If they are a complete orthogonal system, then the inequality (2) becomes an equality, so (6) becomes an equality and is known as Parseval's theorem.

If f(x) has a simple Fourier series expansion with coefficients a_0, a_1, ... , a_n and b_1, ..., b_n, then

The inequality can also be derived from Schwarz's inequality

by expanding g in a superposition of eigenfunctions of f, g=sum_(i)a_if_i. Then

and

where f^_ is the complex conjugate. If g is normalized, then <g|g>=1 and

See also

Complete Orthogonal System, Generalized Fourier Series, Parseval's Theorem, Schwarz's Inequality, Triangle Inequality

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 526-527, 1985.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1102, 2000.Kaplan, W. Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, p. 501, 1992.

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Bessel's Inequality

Cite this as:

Weisstein, Eric W. "Bessel's Inequality." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BesselsInequality.html

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