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 $\{a_n{\}}_{n=1}^{\mathrm{\infty }}\in {\ell }^2$.

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 that $$c\Vert f{\Vert }^2\le \sum _{n=1}^{\mathrm{\infty }}|⟨f,x_n⟩{|}^2\le C\Vert f{\Vert }^2,\mathrm{\forall }f\in H.$$ Hence, Riesz bases need not be orthonormal, i.e., they are a generalization of orthonormal bases.

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

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

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 $H$.

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.

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

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

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

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

The constant 1/4 is sharp; if $\underset{n\in \mathbb{\mathbb{Z}}}{\sup }|{\lambda }_n-n|=1/4$, the sequence may fail to be a Riesz basis, such as:$${\lambda }_n=\{\begin{array}{ll}n-\frac{1}{4},& n>0\\ 0,& n=0\\ n+\frac{1}{4},& n<0\end{array}$$When $\mathrm{\Lambda }=\{{\lambda }_n{\}}_{n\in \mathbb{\mathbb{Z}}}$ are allowed to be complex, the theorem holds under the condition $\underset{n\in \mathbb{\mathbb{Z}}}{\sup }|{\lambda }_n-n|<\frac{\log 2}{\pi }$. 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. 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. 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. https://link.springer.com/content/pdf/bfm:978-0-8176-8224-8/1?pdf=chapter%20toc. 
• 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 (3rd ed.), pp. 46–47, ISBN 9780123743701, https://www.di.ens.fr/~mallat/papiers/WaveletTourChap1-2-3.pdf 
• 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. 
• 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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