Riesz Sequence

In mathematics, a sequence of vectors (''x''_''n'') in a Hilbert space $(H,\langle\cdot,\cdot\rangle)$ is called a Riesz sequence if there exist constants 0 such that

:$c\left( \sum_n , a_n, ^2 \right) \leq \left\Vert \sum_n a_n x_n \right\Vert^2 \leq C \left( \sum_n , a_n, ^2 \right)$

for all sequences of scalars (''a''_''n'') in the ℓ^''p'' space ℓ². A Riesz sequence is called a Riesz basis if

:$\overline = H$ .

Theorems

If ''H'' is a finite-dimensional space, then every basis of ''H'' is a Riesz basis.

Let $\varphi$ be in the ''L''^''p'' space ''L''²(R), let

:$\varphi_n(x) = \varphi(x-n)$

and let $\hat$ denote the Fourier transform of $$. Define constants ''c'' and ''C'' with 0. Then the following are equivalent:

:$1. \quad \forall (a_n) \in \ell^2,\ \ c\left( \sum_n , a_n, ^2 \right) \leq \left\Vert \sum_n a_n \varphi_n \right\Vert^2 \leq C \left( \sum_n , a_n, ^2 \right)$

:$2. \quad c\leq\sum_\left, \hat(\omega + 2\pi n)\^2\leq C$

The first of the above conditions is the definition for ($$) to form a Riesz basis for the space it spans.

See also

* Orthonormal basis
* Hilbert space
* Frame of a vector space

References

*
*

{{PlanetMath attribution, id=7152, title=Riesz basis

Functional analysis