Definitions

Bessel

Bessel

[bes-uhl]
Bessel, Friedrich Wilhelm, 1784-1846, German astronomer and mathematician. He became (1810) director of the new observatory at Königsberg and professor of astronomy at the Univ. of Königsberg. Among his many achievements the most noted is his discovery of the parallax of the fixed star 61 Cygni. Announced in 1838, it was officially recognized in 1841 as the first fully authenticated measurement of the distance of a star. His observations had, by 1833, increased the number of accurately cataloged stars to 50,000. This work was continued and extended by his pupil Argelander. Through observing the variations of the proper motions of Sirius and Procyon, he concluded that they possessed dimmer companions, which was verified a century later by astronomers. Bessel's works on astronomy include Fundamenta Astronomiae (1818) and Astronomische Untersuchungen (1841-42). Bessel also introduced a class of mathematical functions, named for him, which he established as a result of work on perturbation of the planets and which are widely used in applied mathematics, physics, and engineering.
In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence.

Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has

sum_{k=1}^{infty}leftvertleftlangle x,e_krightrangle rightvert^2 le leftVert xrightVert^2

where <∙,∙> denotes the inner product in the Hilbert space H. If we define the infinite sum

x' = sum_{k=1}^{infty}leftlangle x,e_krightrangle e_k,
Bessel's inequality tells us that this series converges.

For a complete orthonormal sequence (that is, for an orthonormal sequence which is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently x' with x).

Bessel's inequality follows from the identity:

left| x - sum_{k=1}^n langle x, e_k rangle e_kright|^2 = |x|^2 - 2 sum_{k=1}^n |langle x, e_k rangle |^2 + sum_{k=1}^n | langle x, e_k rangle |^2 = |x|^2 - sum_{k=1}^n | langle x, e_k rangle |^2,
which holds for any n, excluding when n is less than 1.

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