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 $0<c\leq C<\infty$ such that $$c\sum _{n=1}^{\infty }|a_{n}|^{2}\leq \left\Vert \sum _{n=1}^{\infty }a_{n}x_{n}\right\Vert ^{2}\leq C\sum _{n=1}^{\infty }|a_{n}|^{2},$$ for every finite scalar sequence $\{a_{n}\}$ and hence, for all $\{a_{n}\}_{n=1}^{\infty }\in \ell ^{2}$.

A Riesz sequence is called a Riesz basis if $${\overline {\mathop {\rm {span}} (x_{n})}}=H.$$ Equivalently, a Riesz basis for $H$ is 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. Subsequently, there exist constants $0<c\leq C<\infty$ such that $$c\|f\|^{2}\leq \sum _{n=1}^{\infty }|\langle f,x_{n}\rangle |^{2}\leq C\|f\|^{2},\quad \forall f\in H.$$ Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.

Paley-Wiener criterion

Not to be confused with Paley-Wiener theorem.

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 $H$.

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 $0<c\leq C<+\infty$. 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

See also: Frame (linear algebra) § Non-harmonic Fourier series

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 $L^{2}[-\pi ,\pi ]$.

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:$$\lambda _{n}={\begin{cases}n-{\frac {1}{4}},&n>0\\0,&n=0\\n+{\frac {1}{4}},&n<0\end{cases}}$$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.

See also

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

References

• Antoine, J.-P.; Balazs, P. (2012). "Frames, Semi-Frames, and Hilbert Scales". Numerical Functional Analysis and Optimization. 33 (7–9): 736–769. arXiv:1203.0506. doi:10.1080/01630563.2012.682128. ISSN 0163-0563.
• Balazs, Peter; Stoeva, Diana T.; Antoine, Jean-Pierre (2010). "Classification of General Sequences by Frame-Related Operators". Sampling Theory in Signal and Image Processing. 10 (1–2): 151–170. arXiv:1009.1496. doi:10.1007/BF03549539.
• Christensen, Ole (2016). An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Cham: Springer International Publishing. doi:10.1007/978-3-319-25613-9. ISBN 978-3-319-25611-5.
• Hernandez, Eugenio; Weiss, Guido (1996). A First Course on Wavelets. CRC Press. ISBN 978-0-8493-8274-1.
• Mallat, Stéphane (2008), A Wavelet Tour of Signal Processing: The Sparse Way (PDF) (3rd ed.), pp. 46–47, ISBN 9780123743701
• Paley, Raymond E. A. C.; Wiener, Norbert (1934). Fourier Transforms in the Complex Domain. Providence, RI: American Mathematical Soc. ISBN 978-0-8218-1019-4. {{cite book}}: ISBN / Date incompatibility (help)
• Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. Academic Press. ISBN 978-0-12-772955-8.

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