Hilbert transforms and unitary equivalence

Two-dimensional Hilbert Transforms'

SVD compression, unitary transforms, and computational complexity

The search for fast unitary transforms and the need for data compression in linear systems are complementary issues. Compression requires the definition of a threshold dependent on the condition number, which is invariant over the unitary group. With respect tot this threshold it is shown that the SVD is the optimal tool. Considerations in connection with the Kronecker product and direct sum of...

Multi-Frame Vectors for Unitary Systems in Hilbert $C^{*}$-modules

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 ...

Asymptotically Unitary Equivalence and Asymptotically Inner Automorphisms

Let C be a unital AH-algebra and let A be a unital separable simple C∗-algebra with tracial rank zero. Suppose that φ1, φ2 : C → A are two unital monomorphisms. We show that there is a continuous path of unitaries {ut : t ∈ [0,∞)} of A such that lim t→∞ u∗tφ1(a)ut = φ2(a) for all a ∈ C if and only if [φ1] = [φ2] in KK(C,A), τ ◦ φ1 = τ ◦ φ2 for all τ ∈ T (A) and the rotation map η̃φ1,φ2 associate...

Polynomial Residue Systems via Unitary Transforms

A polynomial, A(z), can be represented by a polynomial residue system and, given enough independent residues, the polynomial can be reconstituted from its residues by the Chinese remainder theorem (CRT). A special case occurs when the discrete Fourier transform and its inverse realise the residue evaluations and CRT respectively, in which case the residue system is realised by the action of a m...