Riesz sequence

In mathematics, a sequence of vectors (x_n) in a Hilbert space $(H,⟨\cdot ,\cdot ⟩)$ is called a Riesz sequence if there exist constants $0<c\le C<+\mathrm{\infty }$ such that

$c(\sum _n|a_n{|}^2)\le {\Vert \sum _n{a}_n{x}_n\Vert }^2\le C(\sum _n|a_n{|}^2)$

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

$\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(x_n)}=H$.

Alternatively, one can define the Riesz basis as a family of the form ${\left\{x_n\right\}}_{n=1}^{\mathrm{\infty }}={\left\{U{e}_n\right\}}_{n=1}^{\mathrm{\infty }}$, where ${\left\{e_n\right\}}_{n=1}^{\mathrm{\infty }}$ is an orthonormal basis for $H$ and $U:H\to H$ is a bounded bijective operator. Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.Template:Sfn

Paley-Wiener criterion

Template:Distinguish Let $\{e_n\}$ be an orthonormal basis for a Hilbert space $H$ and let $\{x_n\}$ be "close" to $\{e_n\}$ in the sense that

$\Vert \sum a_i(e_i-x_i)\Vert \le \lambda \sqrt{\sum |a_i{|}^2}$

for some constant $\lambda$, $0\le \lambda <1$, and arbitrary scalars $a_1,\dots ,a_n$ $(n=1,2,3,\dots )$ . Then $\{x_n\}$ is a Riesz basis for $H$.Template:SfnTemplate:Sfn

Theorems

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

Let $\phi$ be in the L^p space L²(R), let

${\phi }_n(x)=\phi (x-n)$

and let $\hat{\phi }$ denote the Fourier transform of $\phi$. Define constants c and C with $0<c\le C<+\mathrm{\infty }$. Then the following are equivalent:

$1.\mathrm{\forall }(a_n)\in {\ell }^2,c(\sum _n|a_n{|}^2)\le {\Vert \sum _n{a}_n{\phi }_n\Vert }^2\le C(\sum _n|a_n{|}^2)$

$2.c\le \sum _n{|\hat{\phi }(\omega +2\pi n)|}^2\le C$

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

See also

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

References

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