Bessel's inequality explained

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. The inequality was derived by F.W. Bessel in 1828.^[1]

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 ^{inf ty} _{k =1} \left\vert\left\langle x,e _{k\right\rangle} \right\vert ^{2} \le \left\Vert x\right\Vert ^{2,}

where ⟨·,·⟩ denotes the inner product in the Hilbert space

H .^{[2] [3] [4]} If we define the infinite sumx' = \sum ^{inf ty} _{k =1} \left\langle x,e _{k\right\rangle} e _{k,} consisting of "infinite sum" of vector resolute x in direction e _{k} , Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H that can be described in terms of potential basis e _{1,} e _{2,} ... .

For a complete orthonormal sequence (that is, for an orthonormal sequence that 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

\begin{align} 0 \leq \left\| x - \sum ^{n} _{k =1} \langle x, e _{k} \rangle e ^{2} _{k\right\|} & = \|x\| ^{2} - 2 \sum ^{n} _{k =1} \operatorname{Re} \langle x, \langle x, e _{k} \rangle e _{k} \rangle + \sum ^{n} _{k =1} | \langle x, e _{k} \rangle | ^{2} \\ & = \|x\| ^{2} - 2 \sum ^{n} _{k =1} |\langle x, e _{k} \rangle | ^{2} + \sum ^{n} _{k =1} | \langle x, e _{k} \rangle | ^{2} \\ & = \|x\| ^{2} - \sum ^{n} _{k =1} | \langle x, e _{k} \rangle | ^{2, \end{align}} which holds for any natural n.

See also

• Cauchy–Schwarz inequality
• Parseval's theorem

External links

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

Notes and References

1. Web site: Bessel inequality - Encyclopedia of Mathematics.
2. Book: Saxe, Karen. Karen Saxe

   . Beginning Functional Analysis. Karen Saxe . 2001-12-07. Springer Science & Business Media. 9780387952246. 82. en.

3. Book: Mathematical Analysis II. Zorich. Vladimir A.. Cooke. R.. 2004-01-22. Springer Science & Business Media. 9783540406334. 508–509. en.
4. Book: Foundations of Signal Processing. Vetterli. Martin. Kovačević. Jelena. Goyal. Vivek K.. 2014-09-04. Cambridge University Press. 9781139916578. 83. en.