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:

(i)

$\forall c(k)\in l_{2},\ \ A\left\|c\right\|^{2}_{l_{2}}\leq\left\|\sum_{k\in% \mathbb{Z}}c(k)\phi_{k}\right\|^{2}\leq B\left\|c\right\|^{2}_{l_{2}}$
(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
Canonical name	EquivalentConditionForTheTranslatesOfAnL2FunctionToFormARieszSequenceAn
Date of creation	2013-03-22 15:20:07
Last modified on	2013-03-22 15:20:07
Owner	Gorkem (3644)
Last modified by	Gorkem (3644)
Numerical id	15
Author	Gorkem (3644)
Entry type	Theorem
Classification	msc 42C99
Related topic	RieszSequence

equivalent condition for the translates of an L2<math id="m1" class="ltx_Math" alttext="L_{2}" display="inline"><msub><mi>L</mi><mn>2</mn></msub></math> function to form a Riesz sequence, an

Theorem 1

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