Modular frames for Hilbert C*-modules and symmetric approximation of frames

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modulesover unital C*-algebras that admit an orthonormal Riesz basis. Interrelations and applications to classical linear frame theory are indicated. As an application we describe the nature of families of operators {S_i} such that SUM_i S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. Most of the measures are expressed in terms of norm-distances of different kinds of frame operators. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closetst tight frames often contains a multiple of its symmetric orthogonalization (i.e. L\"owdin orthogonalization).Comment: SPIE's Annual Meeting, Session 4119: Wavelets in Signal and Image Processing; San Diego, CA, U.S.A., July 30 - August 4, 2000. to appear in: Proceedings of SPIE v. 4119(2000), 12 p

Framings and dilations

The notion of framings, recently emerging in P. G. Casazza, D. Han, and D. R. Larson, Frames for Banach spaces, in {\em The functional and harmonic analysis of wavelets and frames} (San Antonio, TX, 1999), {\em Contemp. Math}. {\bf 247} (1999), 149-182 as generalization of the reconstraction formula generated by pairs of dual frames, is in this note extended substantially. This calls on refining the basic dilation results which still being in the flavor of {\em th\'eor\`eme principal} of B. Sz-Nagy go much beyond it.Comment: The final version will appear in Acta Sci. Math (Szeged

Irreducible wavelet representations and ergodic automorphisms on solenoids

We focus on the irreducibility of wavelet representations. We present some connections between the following notions: covariant wavelet representations, ergodic shifts on solenoids, fixed points of transfer (Ruelle) operators and solutions of refinement equations. We investigate the irreducibility of the wavelet representations, in particular the representation associated to the Cantor set, introduced in \cite{DuJo06b}, and we present several equivalent formulations of the problem

SNARE VTI13 plays a unique role in endosomal trafficking pathways associated with the vacuole and is essential for cell wall organization and root hair growth in arabidopsis

Background and Aims: Root hairs are responsible for water and nutrient uptake from the soil and their growth is responsive to biotic and abiotic changes in their environment. Root hair expansion is a polarized process requiring secretory and endosomal pathways that deliver and recycle plasma membrane and cell wall material to the growing root hair tip. In this paper, the role of VTI13 (AT3G29100), a member of the VTI vesicular soluble NSF attachment receptor (SNARE) gene family in Arabidopsis thaliana, in root hair growth is described.<p></p> Methods: Genetic analysis and complementation of the vti13 root hair phenotypes of Arabidopsis thaliana were first used to assess the role of VTI13 in root hair growth. Transgenic lines expressing a green fluorescent protein (GFP)–VTI13 construct were used to characterize the intracellular localization of VTI13 in root hairs using confocal microscopy and immunotransmission electron microscopy.<p></p> Key Results: VTI13 was characterized and genetic analysis used to show that its function is required for root hair growth. Expression of a GFP–VTI13 fusion in the vti13 mutant background was shown to complement the vti13 root hair phenotype. GFP–VTI13 localized to both the vacuole membrane and a mobile endosomal compartment. The function of VTI13 was also required for the localization of SYP41 to the trans-Golgi network. Immunohistochemical analysis indicated that cell wall organization is altered in vti13 root hairs and root epidermal cells.<p></p> Conclusions: These results show that VTI13 plays a unique role in endosomal trafficking pathways associated with the vacuole within root hairs and is essential for the maintenance of cell wall organization and root hair growth in arabidopsis

Dilations for Systems of Imprimitivity acting on Banach Spaces

Motivated by a general dilation theory for operator-valued measures, framings and bounded linear maps on operator algebras, we consider the dilation theory of the above objects with special structures. We show that every operator-valued system of imprimitivity has a dilation to a probability spectral system of imprimitivity acting on a Banach space. This completely generalizes a well-kown result which states that every frame representation of a countable group on a Hilbert space is unitarily equivalent to a subrepresentation of the left regular representation of the group. The dilated space in general can not be taken as a Hilbert space. However, it can be taken as a Hilbert space for positive operator valued systems of imprimitivity. We also prove that isometric group representation induced framings on a Banach space can be dilated to unconditional bases with the same structure for a larger Banach space This extends several known results on the dilations of frames induced by unitary group representations on Hilbert spaces.Comment: 21 page