equivalent condition for the translates of an ${L}_{2}$ function to form a Riesz sequence, an
Theorem 1
Let $\varphi \mathrm{\in}{L}_{\mathrm{2}}\mathit{}\mathrm{(}\mathrm{R}\mathrm{)}$, ${\varphi}_{k}\mathit{}\mathrm{(}x\mathrm{)}\mathrm{=}\varphi \mathit{}\mathrm{(}x\mathrm{}k\mathrm{)}$ and $\widehat{\varphi}$ be the Fourier transform^{} of $\varphi $. Let $A$ and $B$ be positive^{} constants. Then the following are equivalent^{}:

(i)
$\forall c(k)\in {l}_{2},A{\parallel c\parallel}_{{l}_{2}}^{2}\le {\parallel {\sum}_{k\in \mathbb{Z}}c(k){\varphi}_{k}\parallel}^{2}\le B{\parallel c\parallel}_{{l}_{2}}^{2}$

(ii)
$A\le {\sum}_{k\in \mathbb{Z}}{\left\widehat{\varphi}(\omega +2\pi k)\right}^{2}\le B$
The first of the above conditions is the definition for ${\{{\varphi}_{k}\}}_{k\in \mathbb{Z}}$ 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  20130322 15:20:07 
Last modified on  20130322 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 