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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\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.^[1]

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 $H$ .^[2]^[3]

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.

Notes

^ Antoine & Balazs 2012.
^ Young 2001, p. 35.
^ Paley & Wiener 1934, p. 100.

References

Antoine, J.-P.; Balazs, P. (2012). "Frames, Semi-Frames, and Hilbert Scales". Numerical Functional Analysis and Optimization. 33 (7–9). arXiv:1203.0506. doi:10.1080/01630563.2012.682128. ISSN 0163-0563.
Christensen, Ole (2001), "Frames, Riesz bases, and Discrete Gabor/Wavelet expansions" (PDF), Bulletin of the American Mathematical Society, New Series, 38 (3): 273–291, doi:10.1090/S0273-0979-01-00903-X
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.
Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition, 93. Academic Press. ISBN 978-0-12-772955-8.

This article incorporates material from Riesz sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. This article incorporates material from Riesz basis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

[FOOTNOTEAntoineBalazs2012-1] Antoine & Balazs 2012.

[FOOTNOTEYoung200135-2] Young 2001, p. 35.

[FOOTNOTEPaleyWiener1934100-3] Paley & Wiener 1934, p. 100.

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Paley-Wiener criterion

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