Bessel's inequality

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.

Countable orthonormal sequence in a Hilbert space

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

$\sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},$

where ⟨·,·⟩ denotes the inner product in the Hilbert space $H$. If we define the infinite sum

$x'=\sum _{k=1}^{\infty }\left\langle x,e_{k}\right\rangle e_{k},$

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 $e_{1},e_{2},\dots$.

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

${\begin{aligned}0\leq \left\|x-\sum _{k=1}^{n}\langle x,e_{k}\rangle e_{k}\right\|^{2}&=\|x\|^{2}-2\sum _{k=1}^{n}\operatorname {Re} \langle x,\langle x,e_{k}\rangle e_{k}\rangle +\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-2\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}+\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2},\end{aligned}}$

which holds for any natural n.

Fourier series

In the theory of Fourier series, in the particular case of the Fourier orthonormal system, we get if $f\colon \mathbb {R} \to \mathbb {C}$ has period $T$,

$\sum _{k\in \mathbb {Z} }\left\vert \int _{0}^{T}e^{-2\pi kt/T}f(t)\,\mathrm {d} t\right\vert ^{2}\leq T\int _{0}^{T}\vert f(t)\vert ^{2}\,\mathrm {d} t.$

In the particular case where $f\colon \mathbb {R} \to \mathbb {R}$, one has then

$\left\vert \int _{0}^{T}f(t)\,\mathrm {d} t\right\vert ^{2}+2\sum _{n=1}^{\infty }\left\vert \int _{0}^{T}\cos(2\pi kt/T)f(t)\,\mathrm {d} t\right\vert ^{2}+2\sum _{n=1}^{\infty }\left\vert \int _{0}^{T}\sin(2\pi kt/T)f(t)\,\mathrm {d} t\right\vert ^{2}\leq T\int _{0}^{T}\vert f(t)\vert ^{2}\,\mathrm {d} t.$

Non countable case

More generally, if $H$ is a pre-Hilbert space and $(e_{\alpha })_{\alpha \in A}$ is an orthonormal system, then for every $x\in H$

$\sum _{\alpha \in A}|\langle x,e_{\alpha }\rangle |^{2}\leq \lVert x\rVert ^{2}$

This is proved by noting that if $F\subseteq A$ is finite, then

$\sum _{\alpha \in F}|\langle x,e_{\alpha }\rangle |^{2}\leq \lVert x\rVert ^{2}$

and then by definition of infinite sum

$\sum _{\alpha \in A}|\langle x,e_{\alpha }\rangle |^{2}={\Bigl \{}\sum _{\alpha \in F}|\langle x,e_{\alpha }\rangle |^{2}:F\subseteq A{\text{ is finite}}{\Bigr \}}\leq \lVert x\rVert ^{2}.$

See also

• Cauchy–Schwarz inequality
• Parseval's theorem

External links

• "Bessel inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Bessel's Inequality the article on Bessel's Inequality on MathWorld.

This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.