Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$

Symmetries in projective multiresolution analyses

We give an equivariant version of Packer and Rieffel's theorem on sufficient conditions for the existence of orthonormal wavelets in projective multiresolution analyses. The scaling functions that generate a projective multiresolution analysis are supposed to be invariant with respect to some finite group action. We give sufficient conditions for the existence of wavelets with similar invariance.

Projective multiresolution analyses for $L^2(R^2)$

We define the notion of "projective" multiresolution analyses, for which, by definition, the initial space corresponds to a finitely generated projective module over the algebra $C(\btn)$ of continuous complex-valued functions on an $n$-torus. The case of ordinary multi-wavelets is that in which the projective module is actually free. We discuss the properties of projective multiresolution analyses, including the frames which they provide for $L^2(\brn)$. Then we show how to construct examples for the case of any diagonal $2 \times 2$ dilation matrix with integer entries, with initial module s…

Projective multiresolution analyses for dilations in higher dimensions

We continue the study of projective module wavelet frames corresponding to diagonal dilation matrices on $\mathbb R^n$ with integer entries, focusing on the construction of a projective multi-resolution analysis corresponding to dilations whose domains are finitely generated projective modules over continuous complex-valued functions on $n$-tori, $n\geq 3$. We are able to generalize some of these results to dilation matrices that are conjugates of integral diagonal dilation matrices by elements of $SL(n,\mathbb Z).$ We follow the method proposed by the author and M. Rieffel, and are able to co…

Wavelet analysis on adeles and pseudo-differential operators

This paper is devoted to wavelet analysis on adele ring $\bA$ and the theory of pseudo-differential operators. We develop the technique which gives the possibility to generalize finite-dimensional results of wavelet analysis to the case of adeles $\bA$ by using infinite tensor products of Hilbert spaces. The adele ring is roughly speaking a subring of the direct product of all possible ($p$-adic and Archimedean) completions $\bQ_p$ of the field of rational numbers $\bQ$ with some conditions at infinity. Using our technique, we prove that $L^2(\bA)=\otimes_{e,p\in\{\infty,2,3,5,...}}L^2({\bQ}_{…

非交换多项式的类Gram矩阵保持扩展与完备化

Gram-like matrix preserving extensions and completions of noncommutative polynomials

针对非交换多项式扩展为埃尔米特平方和时无法保持Gram矩阵不变的问题，本文提出类Gram矩阵概念及线性代数技术实现扩展，并利用半正定完备化问题结合弦图与2-正则射影代数集给出判据。