Frames of translates with prescribed fine structure in shift invariant spaces

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• 作者：Marí ; a J. Benac¹ ; ^{mjbenac@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Pedro G. Massey¹ ; ^{massey@mate.unlp.edu.ar" class="auth_mail" title="E-mail the corresponding author} ; Demetrio Stojanoff¹ ; ^{demetrio@mate.unlp.edu.ar" class="auth_mail" title="E-mail the corresponding author}
• 关键词：42C15
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 November 2016
• 年：2016
• 卷：271
• 期：9
• 页码：2631-2671
• 全文大小：682 K

文摘

For a given finitely generated shift invariant (FSI) subspace rc">rce">W⊂L²(R^k)$\mathcal{W}\subset L^2({\mathbb{R}}^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences rc">rce">E(F)$E(\mathcal{F})$ induced by finite sequences of vectors rc">rce">F∈Wⁿ$\mathcal{F}\in {\mathcal{W}}^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in rc">rce">F$\mathcal{F}$ and the spectra of rc">rce">S_E(F)$S_{E(\mathcal{F})}$ are prescribed in each fiber of rc">rce">Spec(W)⊂T^k$\mathrm{Spec}(\mathcal{W})\subset {\mathbb{T}}^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given rc">rce">α₁≥…≥α_n>0${\alpha }_1\ge \dots \ge {\alpha }_n>0$ we characterize the finite sequences rc">rce">F∈Wⁿ$\mathcal{F}\in {\mathcal{W}}^n$ such that rc">rce">‖f_i‖²=α_i${\Vert f_i\Vert }^2={\alpha }_i$, for rc">rce">1≤i≤n$1\le i\le n$, and such that the fine spectral structure of the shift generated Bessel sequences rc">rce">E(F)$E(\mathcal{F})$ has minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential rc">rce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616301951&_mathId=si10.gif&_user=111111111&_pii=S0022123616301951&_rdoc=1&_issn=00221236&md5=0b91b5b6d130e6f632890f785810f854">rce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616301951-si10.gif">$P_{\phi }^{\mathcal{W}}$ induced by rc">rce">W$\mathcal{W}$ and an arbitrary convex function rc">rce">φ:R₊→R₊$\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$.