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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.

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Licensed from Columbia University Press

Licensed from Columbia University Press

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.## External links

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

- $x\text{'}\; =\; sum\_\{k=1\}^\{infty\}leftlangle\; x,e\_krightrangle\; e\_k,$

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 $x\text{'}$ with $x$).

Bessel's inequality follows from the identity:

- $left|\; x\; -\; sum\_\{k=1\}^n\; langle\; x,\; e\_k\; rangle\; e\_kright|^2\; =\; |x|^2\; -\; 2\; sum\_\{k=1\}^n\; |langle\; x,\; e\_k\; rangle\; |^2\; +\; sum\_\{k=1\}^n\; |\; langle\; x,\; e\_k\; rangle\; |^2\; =\; |x|^2\; -\; sum\_\{k=1\}^n\; |\; langle\; x,\; e\_k\; rangle\; |^2$,

- Bessel's Inequality the article on Bessel's Inequality on MathWorld.

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Last updated on Friday September 26, 2008 at 22:23:24 PDT (GMT -0700)

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