In this paper, we focus on the structured multi-frame vectors in Hilbert $C^*$-modules. More precisely, it will be shown that the set of all complete multi-frame vectors for a unitary system can be parameterized by the set of all surjective operators, in the local commutant. Similar results hold for the set of all complete wandering vectors and complete multi-Riesz vectors, when the surjective ...

In this paper, we obtain equivalent conditions relating oblique projections to biorthogonal Riesz bases and angles between closed linear subspaces of a Hilbert space. We also prove an extension theorem in the biorthogonal setting, which leads to biorthogonal multiwavelets.

We introduce the basic concepts of Gröbner basis theory and its relations to polytope theory. This will cover very basic parts of [1], the basic chapters from [3] and the first chapters from [11]. In particular, we will define the Gröbner fan, which will play a major role in some of the research problems we will pose later. In this lecture we relate triangulations to Gröbner bases and Hilbert-s...

Normaliz is a program for the computation of Hilbert bases of rational cones and the normalizations of affine monoids. It may also be used for solving diophantine linear systems. In this paper we present the algorithms implemented in the program.

By introducing Hilbert space and operators, we show how probabilities, approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates. Our main results yield orthogonal bases which optimize distinct measures of data encoding.

This paper deals with density and regression estimation problems for functional data. Using wavelet bases for Hilbert spaces of functions, we develop a new adaptive procedure based on wavelet thresholding. We provide theoretical results on its asymptotic performances.

Given a self-adjoint operator A in Hilbert space H, we analyze its spectral behavior when it is expressed terms of generalized eigenvectors. Using the formalism Gel’fand distribution bases, explore conditions for eigenspaces to be one-dimensional, i.e., have simple spectrum.

We consider two quantum cryptographic schemes relying on encoding the key into qudits, i.e., quantum states in a d-dimensional Hilbert space. The first cryptosystem uses two mutually unbiased bases (thereby extending the BB84 scheme), while the second exploits all d+1 available such bases (extending the six-state protocol for qubits). We derive the information gained by a potential eavesdropper...