In mathematics, a sequence of vectors (xn) in a Hilbert space
is called a Riesz sequence if there exist constants
such that
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1992f45b6fece33130468c4ac4c9a5239e2e8)
for all sequences of scalars (an) in the ℓp space ℓ2. A Riesz sequence is called a Riesz basis if
.
Alternatively, one can define the Riesz basis as a family of the form
, where
is an orthonormal basis for
and
is a bounded bijective operator.
Paley-Wiener criterion
Let
be an orthonormal basis for a Hilbert space
and let
be "close" to
in the sense that
![{\displaystyle \left\|\sum a_{i}(e_{i}-x_{i})\right\|\leq \lambda {\sqrt {\sum |a_{i}|^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d875a4aa2dcab5a2def08420ad0ac752583f1f38)
for some constant
,
, and arbitrary scalars
. Then
is a Riesz basis for
. 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
be in the Lp space L2(R), let
![{\displaystyle \varphi _{n}(x)=\varphi (x-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86bb635ff5ff9deb71e4c9dd1613ef8352ae21f8)
and let
denote the Fourier transform of
. Define constants c and C with
. Then the following are equivalent:
![{\displaystyle 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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80ff0065b10d3f26adfc19f2f4cd5d86b5d0196a)
![{\displaystyle 2.\quad c\leq \sum _{n}\left|{\hat {\varphi }}(\omega +2\pi n)\right|^{2}\leq C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b412cc84e2c21df98f2c446035a92b09260ab1b)
The first of the above conditions is the definition for (
) to form a Riesz basis for the space it spans.
See also
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
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.