Riesz sequence

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In mathematics, a sequence of vectors (x_n) in a Hilbert space (H, 〈 , 〉) 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

$\overline{\mathop{\rm span} (x_n)} = H$ .

[edit] Theorems

If H is a finite-dimensional space, then every basis of H is a Riesz basis.

Let φ be in the L^p space L²(R), let

φ n (x) = φ(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:

$1. \quad \forall (a_n) \in \ell^2,\ \ c\left( \sum_n | a_n|^2 \right) \leq \left\Vert \sum_n a_n \varphi_n \right\Vert^2 \leq C \left( \sum_n | a_n|^2 \right)$