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
:$\sum_^\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2,$

where ⟨·,·⟩ denotes the inner product in the Hilbert space $H$. If we define the infinite sum
:$x' = \sum_^\left\langle x,e_k\right\rangle e_k,$
consisting of the "infinite sum" of the vector resolute $x$ in the directions $e_k$, 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
:$\begin 0 \leq \left\, x - \sum_^n \langle x, e_k \rangle e_k\right\, ^2 &= \, x\, ^2 - 2 \sum_^n \operatorname \langle x, \langle x, e_k \rangle e_k \rangle + \sum_^n , \langle x, e_k \rangle , ^2 \\ &= \, x\, ^2 - 2 \sum_^n , \langle x, e_k \rangle , ^2 + \sum_^n , \langle x, e_k \rangle , ^2 \\ &= \, x\, ^2 - \sum_^n , \langle x, e_k \rangle , ^2, \end$
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 $f \colon \mathbb \to \mathbb$ has period $T$,
:$\sum_ \left\vert \int_0^T e^ f (t) \,\mathrmt\right\vert^2 \le T \int_0^T \vert f (t)\vert^2 \,\mathrmt.$
In the particular case where $f \colon \mathbb \to \mathbb$, one has then
:$\left\vert \int_0^T f (t) \,\mathrmt\right\vert^2 + 2 \sum_^\infty \left\vert \int_0^T \cos (2 \pi k t/T) f (t) \,\mathrmt\right\vert^2 + 2 \sum_^\infty \left\vert \int_0^T \sin (2 \pi k t/T) f (t) \,\mathrmt\right\vert^2 \le T \int_0^T \vert f (t)\vert^2 \,\mathrmt.$

Non countable case

More generally, if $H$ is a pre-Hilbert space and $(e_\alpha)_$ is an orthonormal system, then for every $x \in H$
:$\sum_ , \langle x, e_\alpha \rangle , ^2 \le \lVert x \rVert^2$
This is proved by noting that if $F \subseteq A$ is finite, then
:$\sum_ , \langle x, e_\alpha \rangle , ^2 \le \lVert x \rVert^2$
and then by definition of infinite sum
:$\sum_ , \langle x, e_\alpha \rangle , ^2 = \sup \Bigl\ \le \lVert x \rVert^2.$

See also

* Cauchy–Schwarz inequality
* Parseval's theorem

References

External links

*

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

{{Hilbert space

Hilbert spaces
Inequalities (mathematics)