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 ${\varphi }$ . 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 ( ${\varphi _{n}}$ ) 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

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Riesz sequence

Theorems

See also

References