# equivalent condition for the translates of an $L_{2}$ function to form a Riesz sequence, an

###### Theorem 1

Let $\phi\in L_{2}(\mathbb{R})$, $\phi_{k}(x)=\phi(x-k)$ and $\hat{\phi}$ be the Fourier transform of $\phi$. Let $A$ and $B$ be positive constants. Then the following are equivalent:

1. (i)

$\forall c(k)\in l_{2},\ \ A\left\|c\right\|^{2}_{l_{2}}\leq\left\|\sum_{k\in% \mathbb{Z}}c(k)\phi_{k}\right\|^{2}\leq B\left\|c\right\|^{2}_{l_{2}}$

2. (ii)

$A\leq\sum_{k\in\mathbb{Z}}\left|\hat{\phi}(\omega+2\pi k)\right|^{2}\leq B$

The first of the above conditions is the definition for $\{\phi_{k}\}_{k\in\mathbb{Z}}$ to form a Riesz sequence.

Title equivalent condition for the translates of an $L_{2}$ function to form a Riesz sequence, an EquivalentConditionForTheTranslatesOfAnL2FunctionToFormARieszSequenceAn 2013-03-22 15:20:07 2013-03-22 15:20:07 Gorkem (3644) Gorkem (3644) 15 Gorkem (3644) Theorem msc 42C99 RieszSequence