BesselUngleichung

The Bessel inequality is a fundamental result in the theory of Hilbert spaces, named after the German mathematician Friedrich Bessel. It establishes an upper bound for the sum of the squares of the coefficients of the Fourier series of a function with respect to an orthonormal system.

In a Hilbert space H with an orthonormal system {e_i}, for any vector x in H, the Bessel inequality states that the sum of the absolute squares of the inner products of x with each orthonormal vector e_i is less than or equal to the square of the norm of x. Mathematically, this is expressed as:

Sum_{i=1 to infinity} |<x, e_i>|^2 <= ||x||^2

where <x, e_i> denotes the inner product of x and e_i, and ||x|| denotes the norm of x. The inequality holds for any orthonormal system, which may be finite or infinite.

A key implication of the Bessel inequality is that it ensures the convergence of the series of squared Fourier coefficients. This is crucial for understanding the properties of functions in Hilbert spaces and for developing approximation methods. When the equality holds in the Bessel inequality for a given vector x and an orthonormal system {e_i}, it signifies that the orthonormal system is complete, meaning that x can be perfectly represented as a linear combination of the vectors in the system. This is known as Parseval's identity. The Bessel inequality is a foundational concept in Fourier analysis, signal processing, and quantum mechanics.