In mathematics, a sequence of vectors (x_n) in a Hilbert space ${\displaystyle (H,\langle \cdot ,\cdot \rangle )}$ is called a Riesz sequence if there exist constants ${\displaystyle 0<c\leq C<+\infty }$ such that

${\displaystyle c\left(\sum _{n}|a_{n}|^{2}\right)\leq \left\Vert \sum _{n}a_{n}x_{n}\right\Vert ^{2}\leq C\left(\sum _{n}|a_{n}|^{2}\right)}$

for all sequences of scalars (a_n) in the ℓ^p space ℓ². A Riesz sequence is called a Riesz basis if

${\displaystyle {\overline {\mathop {\rm {span}} (x_{n})}}=H}$.

Alternatively, one can define the Riesz basis as a family of the form ${\displaystyle \left\{x_{n}\right\}_{n=1}^{\infty }=\left\{Ue_{n}\right\}_{n=1}^{\infty }}$, where ${\displaystyle \left\{e_{n}\right\}_{n=1}^{\infty }}$ is an orthonormal basis for ${\displaystyle H}$ and ${\displaystyle U:H\rightarrow H}$ is a bounded bijective operator.

Paley-Wiener criterion

Let ${\displaystyle \{e_{n}\}}$ be an orthonormal basis for a Hilbert space ${\displaystyle H}$ and let ${\displaystyle \{x_{n}\}}$ be "close" to ${\displaystyle \{e_{n}\}}$ in the sense that

${\displaystyle \left\|\sum a_{i}(e_{i}-x_{i})\right\|\leq \lambda {\sqrt {\sum |a_{i}|^{2}}}}$

for some constant ${\displaystyle \lambda }$, ${\displaystyle 0\leq \lambda <1}$, and arbitrary scalars ${\displaystyle a_{1},\dotsc ,a_{n}}$ ${\displaystyle (n=1,2,3,\dotsc )}$ . Then ${\displaystyle \{x_{n}\}}$ is a Riesz basis for ${\displaystyle H}$. 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 ${\displaystyle \varphi }$ be in the L^p space L²(R), let

${\displaystyle \varphi _{n}(x)=\varphi (x-n)}$

and let ${\displaystyle {\hat {\varphi }}}$ denote the Fourier transform of ${\displaystyle {\varphi }}$. Define constants c and C with ${\displaystyle 0<c\leq C<+\infty }$. Then the following are equivalent:

${\displaystyle 1.\quad \forall (a_{n})\in \ell ^{2},\ \ c\left(\sum _{n}|a_{n}|^{2}\right)\leq \left\Vert \sum _{n}a_{n}\varphi _{n}\right\Vert ^{2}\leq C\left(\sum _{n}|a_{n}|^{2}\right)}$

${\displaystyle 2.\quad c\leq \sum _{n}\left|{\hat {\varphi }}(\omega +2\pi n)\right|^{2}\leq C}$

The first of the above conditions is the definition for (${\displaystyle {\varphi _{n}}}$) to form a Riesz basis for the space it spans.

See also

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

References

• 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.