arXiv subjects
Publications and source records attributed to Marcin Bownik.
We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.
In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions less than ten.
We extend an estimate of Taibleson and Weiss, regarding Fourier transform of Hardy spaces, to the aniostropic setting. As consequences, we obtain necessary conditions for multiplier operators, and the anisotropic version of the Hardy-Littlewood inequality. This last inequality is strengthened with a rearrangement argument.
Let $A_i$ for $i=1, 2$ be an expansive dilation, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$ and $\vec A\equiv(A_1, A_2)$. Denote by ${\mathcal A}_\infty(\rnm; \vec A)$ the class of Muckenhoupt weights associated with $\vec A$. The authors introduce a class of anisotropic singular integrals on $\mathbb R^n\times\mathbb R^m$, whose kernels are adapted to $\vec A$ in the sense of Bownik and have vanishing moments defined via bump functions in the sense of Stein. Then the authors establish the boundedness of these anisotropic singular integrals on $L^q_w(\mathbb R^n\times\mathbb R^m)$ with $q\in(1, \infty)$ and $w\in\mathcal A_q(\mathbb R^n\times\mathbb R^m; \vec A)$ or on $H^p_w(\mathbb R^n\times\mathbb R^m; \vec A)$ with $p\in(0, 1]$ and $w\in\mathcal A_\infty(\mathbb R^n \times\mathbb R^m; \vec A)$. These results are also new even when $w=1$.
We establish dilation theorems for non-tight frames with additional structure, i.e., frames generated by unitary groups of operators and projective unitary representations. This generalizes previous dilation results for Parseval frames due to Han and Larson and Gabardo and Han. We also extend the dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun , by identifying the optimal class of frame wavelets for which dilation into an orthonormal wavelet is possible.
Let $A_1$ and $A_2$ be expansive dilations, respectively, on ${\mathbb R}^n$ and ${\mathbb R}^m$. Let $\vec A\equiv(A_1, A_2)$ and $\mathcal A_p(\vec A)$ be the class of product Muckenhoupt weights on ${\mathbb R}^n\times{\mathbb R}^m$ for $p\in(1, \infty]$. When $p\in(1, \infty)$ and $w\in{\mathcal A}_p(\vec A)$, the authors characterize the weighted Lebesgue space $L^p_w({\mathbb R}^n\times{\mathbb R}^m)$ via the anisotropic Lusin-area function associated with $\vec A$. When $p\in(0, 1]$, $w\in {\mathcal A}_\infty(\vec A)$, the authors introduce the weighted anisotropic product Hardy space $H^p_w({\mathbb R}^n\times{\mathbb R}^m; \vec A)$ via the anisotropic Lusin-area function and establish its atomic decomposition. Moreover, the authors prove that finite atomic norm on a dense subspace of $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ is equivalent with the standard infinite atomic decomposition norm. As an application, the authors prove that if $T$ is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $\mathcal B $, then $T$ uniquely extends to a bounded sublinear operator from $H^p_w({\mathbb R}^n\times{\mathbb R}^m;\vec A)$ to $\mathcal B$. The results of this paper improve the existing results for weighted product Hardy spaces and are new even in the unweighted anisotropic setting.