Riesz sequence

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

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

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 (New Series) of the American Mathematical Society, 38 (3): 273–291
• Mallat, Stéphane (2008), A Wavelet Tour of Signal Processing: The Sparse Way (PDF) (3th 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.