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 $$c\sum _{n=1}^{\mathrm{\infty }}|a_n{|}^2\le {\Vert \sum _{n=1}^{\mathrm{\infty }}{a}_n{x}_n\Vert }^2\le C\sum _{n=1}^{\mathrm{\infty }}|a_n{|}^2,$$ for every finite scalar sequence $\{a_n\}$ and hence, for all $\{a_n{\}}_{n=1}^{\mathrm{\infty }}\in {\ell }^2$.Template:SfnTemplate:Sfn

A Riesz sequence is called a Riesz basis if $$\overline{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}(x_n)}=H.$$ Equivalently, a Riesz basis for $H$ is 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. Subsequently, there exist constants $0<c\le C<\mathrm{\infty }$ such thatTemplate:Sfn $$c\Vert f{\Vert }^2\le \sum _{n=1}^{\mathrm{\infty }}|⟨f,x_n⟩{|}^2\le C\Vert f{\Vert }^2,\mathrm{\forall }f\in H.$$ 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:Template:Sfn

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

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

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

Kadec 1/4 Theorem

Template:See also The Kadec 1/4 theorem, sometimes called the Kadets 1/4 theorem, provides a specific condition under which a sequence of complex exponentials forms a Riesz basis for the Lp space $L^2[-\pi ,\pi ]$. It is a foundational result in the theory of non-harmonic Fourier series.

Let $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n\in \mathbb{Z}}$ be a sequence of real numbers such that

$\underset{n\in \mathbb{Z}}{sup}|{\lambda }_n-n|<\frac{1}{4}$

Then the sequence of complex exponentials $\{e^{i{\lambda }_{n}t}{\}}_{n\in \mathbb{Z}}$ forms a Riesz basis for $L^2[-\pi ,\pi ]$.Template:Sfn

This theorem demonstrates the stability of the standard orthonormal basis $\{e^{int}{\}}_{n\in \mathbb{Z}}$ (up to normalization) under perturbations of the frequencies $n$.

The constant 1/4 is sharp; if $\underset{n\in \mathbb{Z}}{sup}|{\lambda }_n-n|=1/4$, the sequence may fail to be a Riesz basis, such as:Template:Sfn$${\lambda }_n=\{\begin{array}{cc}n-\frac{1}{4},& n>0\\ 0,& n=0\\ n+\frac{1}{4},& n<0\end{array}$$When $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n\in \mathbb{Z}}$ are allowed to be complex, the theorem holds under the condition $\underset{n\in \mathbb{Z}}{sup}|{\lambda }_n-n|<\frac{\log 2}{\pi }$. Whether the constant is sharp is an open question.Template:Sfn

See also

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

References

• Template:Cite journal
• Template:Cite journal
• Template:Cite book
• Template:Cite book
• Template:Citation
• Template:Cite book
• Template:Cite book