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

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 φ. 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 (φ_n) to form a Riesz basis for the space it spans.

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

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.

This article is issued from Wikipedia - version of the 10/15/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Riesz sequence

Theorems

See also

References