A Multidimensional Critical Factorization Theorem

The Critical Factorization Theorem is one of the principal results in combinatorics on words. It relates local periodicities of a word to its global periodicity. In this paper we give a multidimensional extension of it. More precisely, we give a new proof of the Critical Factorization Theorem, but in a weak form, where the weakness is due to the fact that we loose the tightness of the local repetition order. In exchange, we gain the possibility of extending our proof to the multidimensional case. Indeed, this new proof makes use of the Theorem of Fine and Wilf, that has several classical generalizations to the multidimensional cas

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for constructing 2D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1D counterpart, we relate the real and imaginary components of these complex wavelets using a multi-dimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient FFT-based filterbank algorithm for implementing the associated complex wavelet transform.Comment: 36 pages, 8 figure

On Characterization of Inverse Data in the Boundary Control Method

We deal with a dynamical system \begin{align*} & u_{tt}-\Delta u+qu=0 && {\rm in}\,\,\,\Omega \times (0,T)\\ & u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,\,\overline \Omega\\ & \partial_\nu u = f && {\rm in}\,\,\,\partial\Omega \times [0,T]\,, \end{align*} where $\Omega \subset {\mathbb R}^n$ is a bounded domain, $q \in L_\infty(\Omega)$ a real-valued function, $\nu$ the outward normal to $\partial \Omega$, $u=u^f(x,t)$ a solution. The input/output correspondence is realized by a response operator $R^T: f \mapsto u^f\big|_{\partial\Omega \times [0,T]}$ and its relevant extension by hyperbolicity $R^{2T}$. Ope\-rator $R^{2T}$ is determined by $q\big|_{\Omega^T}$, where $\Omega^T:=\{x \in \Omega\,|\,\,{\rm dist\,}(x,\partial \Omega)<T\}$. The inverse problem is: Given $R^{2T}$ to recover $q$ in $\Omega^T$. We solve this problem by the boundary control method and describe the {\it ne\-ces\-sary and sufficient} conditions on $R^{2T}$, which provide its solvability.Comment: 33 pages, 1 figur