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\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.

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 $H$ . Hence, Riesz bases need not be orthonormal.[1]

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.

References

Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition. Academic Press. p. 35. ISBN 978-0-12-772955-8.

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

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[1] Young, Robert M. (2001). An Introduction to Non-Harmonic Fourier Series, Revised Edition. Academic Press. p. 35. ISBN 978-0-12-772955-8.

Riesz sequence

Paley-Wiener criterion

Theorems

See also

References