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.

单词	EquivalentConditionForTheTranslatesOfAnL2FunctionToFormARieszSequenceAn
释义	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.
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equivalent condition for the translates of an L2 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