Spaces Generated by Orbits of Unitary Representations: A Tribute to Guido Weiss (original) (raw)

Invariant spaces under unitary representations of discrete groups

arXiv: Functional Analysis, 2018

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using the Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to countable families of generators previous results obtained for principal subspaces.

Spaces invariant under unitary representations of discrete groups

Journal of Mathematical Analysis and Applications, 2020

We investigate the structure of subspaces of a Hilbert space that are invariant under unitary representations of a discrete group. We work with square integrable representations, and we show that they are those for which we can construct an isometry intertwining the representation with the right regular representation, that we call a Helson map. We then characterize invariant subspaces using a Helson map, and provide general characterizations of Riesz and frame sequences of orbits. These results extend to the nonabelian setting several known results for abelian groups. They also extend to countable families of generators previous results obtained for principal subspaces.

Unitary representations of wavelet groups and encoding of iterated function systems in solenoids

Ergodic Theory and Dynamical Systems, 2009

For points in d real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed d by d matrix over Z. Our starting point is a given pair (A, D) with the matrix A assumed expansive, and D a chosen complete digit set, i.e., in bijective correspondence with the points in Z d /A T Z d . We give an explicit geometric representation and encoding with infinite words in letters from D. We show that the attractor X(A T , D) for an affine Iterated Function System (IFS) based on (A, D) is a set of fractions for our digital representation of points in R d . Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid S A associated with the pair . The intricate part (Theorem 6.15) is played by the cycles in Z d for the initial (A, D)-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional D-representation.

Wavelet subspaces invariant under groups of translation operators

Proceedings Mathematical Sciences, 2003

We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.

Wavelet filters and infinite-dimensional unitary groups

2000

In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C^*-algebra O_N. A main tool in our analysis is the infinite-dimensional group of all maps T -> U(N) (where U(N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.

On the structure of the space of wavelet transforms

Comptes Rendus Mathematique, 2008

Let G be the "ax + b"-group with the left invariant Haar measure dν and ψ be a fixed real-valued admissible wavelet on L2(R). The complete decomposition of L2(G, dν) onto the space of wavelet transforms W ψ (L2(R)) is obtained after identifying the group G with the upper half-plane Π in C. To cite this article: O. Hutník, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The Structure of Finitely Generated Shift-Invariant Spaces in

1992

A simple characterization is given of finitely generated subspaces of L2(Rd) which are invariant under translation by any (multi)integer, and is used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for "local" spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces. As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years′ standing.

Fe b 20 00 Wavelet filters and infinite-dimensional unitary groups

2000

Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C∗-algebra ON . A main tool in our analysis is the infinite-dimensional group of all maps T → U (N) (where U (N) is the group of all unitary N-by-N matrices), and we study the extension problem from low-pass filter to multiresolution filter using this group.

On the (U+K)-orbits of certain weighted shifts

Integral Equations and Operator Theory, 1993

has defined the (/4 +/(:)-orbit of an operator T acting on an nilbert space as (U+/C)(T) = {R-1TR : R invertible of the form unitary plus compact}. In this paper we consider the (U +/C)-orbit and the closure thereof for bilateral and unilateral weighted shifts. In particular, we determine which shifts are in the (//+/C)-orbits of injective weighted shifts and which shifts are in the closure of the (/d +/(:)-orbit of periodic injective shifts. Also, the closure of the (U +/(:)-orbit of injective essentially normal shifts is determined.