Riesz sequence#Paley-Wiener criterion

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

: $\overline{\mathop{\rm span} (x_n)} = H$ .

Alternatively, one can define the Riesz basis as a family of the form $\left\{x_{n} \right\}_{n=1}^{\infty} = \left\{ Ue_{n} \right\}_{n=1}^{\infty}$ , where $\left\{e_{n} \right\}_{n=1}^{\infty}$ is an orthonormal basis for $H$ and $U : H \rightarrow H$ is a bounded bijective operator. Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.{{sfn | Antoine | Balazs | 2012}}

Paley-Wiener criterion

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

: $\left\| \sum a_{i} (e_{i} - x_{i})\right\| \leq \lambda \sqrt{\sum |a_{i}|^{2}}$

for some constant $\lambda$ , $0 \leq \lambda < 1$ , and arbitrary scalars $a_{1},\dotsc, a_{n}$ $(n = 1,2,3,\dotsc)$ . Then $\{x_{n}\}$ is a Riesz basis for $H$ .{{sfn | Young | 2001 | p=35}}{{sfn | Paley | Wiener | 1934 | p=100}}

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 L²(R), let

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

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

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

: $2. \quad c\leq\sum_{n}\left|\hat{\varphi}(\omega + 2\pi n)\right|^2\leq C$

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

Kadec 1/4 Theorem

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 $\Lambda = \{\lambda_n\}_{n \in \mathbb{Z}}$ be a sequence of real numbers such that

: $\sup_{n \in \mathbb{Z}} |\lambda_n - n| < \frac{1}{4}$

Then the sequence of complex exponentials $\{e^{i \lambda_n t}\}_{n \in \mathbb{Z}}$ forms a Riesz basis for $L^2[-\pi, \pi]$ .{{sfn | Young | 2001 | p=36}}

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 $\sup_{n \in \mathbb{Z}} |\lambda_n - n| = 1/4$ , the sequence may fail to be a Riesz basis, such as:{{sfn|Young|2001|p=37}} $\lambda_n= \begin{cases}n-\frac{1}{4}, & n>0 \\ 0, & n=0 \\ n+\frac{1}{4}, & n<0\end{cases}$ When $\Lambda = \{\lambda_n\}_{n \in \mathbb{Z}}$ are allowed to be complex, the theorem holds under the condition $\sup_{n \in \mathbb{Z}} |\lambda_n - n| < \frac{\log 2}{\pi}$ . Whether the constant is sharp is an open question.{{sfn|Young|2001|p=37}}

Notes

References

{{cite journal | last=Antoine | first=J.-P. | last2=Balazs | first2=P. | title=Frames, Semi-Frames, and Hilbert Scales | journal=Numerical Functional Analysis and Optimization | volume=33 | issue=7-9 | date=2012 | issn=0163-0563 | doi=10.1080/01630563.2012.682128| arxiv=1203.0506 }}
{{Citation | last1=Christensen | first1=Ole | title=Frames, Riesz bases, and Discrete Gabor/Wavelet expansions | url=https://www.ams.org/journals/bull/2001-38-03/S0273-0979-01-00903-X/S0273-0979-01-00903-X.pdf | year=2001 | journal=Bulletin of the American Mathematical Society |series=New Series | volume=38 | issue=3 | pages=273–291 | doi=10.1090/S0273-0979-01-00903-X | doi-access=free }}
{{Citation | last=Mallat | first=Stéphane | title=A Wavelet Tour of Signal Processing: The Sparse Way | url=https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf | year=2008 | isbn=9780123743701 | edition = 3rd | pages=46–47 }}
{{cite book |authorlink1=Raymond Paley | authorlink2=Norbert Wiener| last=Paley | first=Raymond E. A. C. | last2=Wiener | first2=Norbert | title=Fourier Transforms in the Complex Domain | publisher=American Mathematical Soc. | publication-place=Providence, RI | date=1934 | isbn=978-0-8218-1019-4}}
{{cite book | last=Young | first=Robert M. | title=An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93 | publisher=Academic Press | date=2001 | isbn=978-0-12-772955-8}}

Category:Functional analysis