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 $0<c\le C<+\infty$ 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 $0<c\le C<+\infty$. 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.

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

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.