小波系统生成元集的稠密性
Density of the Set of Generators of Wavelet Systems
摘要：
给定$ \LL^2(\R^d)$中的函数$ψ$，由$ψ$生成的与可逆矩阵$a$和格$\zG$相关联的仿射（小波）系统是函数$\{|\det a|^{j/2} ψ(a^jx-γ): j \in \Z, γ\in \zG\}$的集合。在本文中，我们证明了生成仿射系统的函数集（这些仿射系统是$ \LL^2(\R^d)$的Riesz基）在$ \LL^2(\R^d)$中是稠密的。我们还证明了一个更强的结果是真实的仿射系统是一个框架的$ \LL^2(\R^d)$。在这种情况下，我们表明，与一个固定的，但任意膨胀的发电机是一个稠密的。此外，我们分析了正交的情况下，我们证明了正交（不一定是完整的）仿射系统的生成元的集合，这是紧支持的频率，是密集的单位球$ \LL^2(\R^d)$与诱导度量。作为一个副产品，我们介绍了$p$ -Grammian的功能，并证明了收敛结果，这Grammian作为一个功能的格。这一结果提供了洞察仿射系统的过采样问题。
英文摘要：
Given a function $ψ$ in $ \LL^2(\R^d)$, the affine (wavelet) system generated by $ψ$, associated to an invertible matrix $a$ and a lattice $\zG$, is the collection of functions $\{|\det a|^{j/2} ψ(a^jx-γ): j \in \Z, γ\in \zG\}$. In this article we prove that the set of functions generating affine systems that are a Riesz basis of $ \LL^2(\R^d)$ is dense in $ \LL^2(\R^d)$. We also prove that a stronger result is true for affine systems that are a frame of $ \LL^2(\R^d)$. In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems, that are compactly supported in frequency, are dense in the unit sphere of $ \LL^2(\R^d)$ with the induced metric. As a byproduct we introduce the $p$-Grammian of a function and prove a convergence result of this Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems.
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