Riesz sequence

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A sequence of vectors $(x n)$ in a Hilbert space $H$ is called a Riesz sequence if there exist constants $0 < c \leq C$ 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 $(a_n) \in l^2$ . 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 $\phi \in L_2(\mathbb{R})$ , $φ n (x) = φ(x - n)$ and $\hat{\phi}$ be the Fourier transform of $φ$ . Define constants $c$ and $C$ sch that $0 < c \leq C$ . Then the following are equivalent:

$1. \quad \forall (a_n) \in l_2,\ \ c\left( \sum_n | a_n|^2 \right) \leq \left\Vert \sum_n a_n \phi_n \right\Vert^2 \leq C \left( \sum_n | a_n|^2 \right)$