Wai-Shing Tang

Wavelets in wandering subspaces

Mallats construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite B-splines is constructed

Oblique projections, biorthogonal Riesz bases and multiwavelets in Hilbert spaces

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.

On positive linear maps between matrix algebras

Abstract There is a concrete example of a positive linear map from M 2 to M 4 which is not decomposable. Modification of this map gives an explicit counterexample to a conjecture of Woronowicz on the strong Kadison inequality.

Geometric aspects of frame representations of abelian groups

hhWe consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

Wavelet bases for a set of commuting unitary operators

AbstractLet (U=U1, ...,Ud) be an orderedd-tuple of distinct, pairwise commuting, unitary operators on a complex Hilbert space ℋ, and letX:={x1, ...,xr} ⊂ ℋ such that

Oblique multiwavelets in Hilbert spaces

U^{\mathbb{Z}^d } X: = \{ U_1^{n_1 } \ldots U_d^{n_d } x_j :(n_1 , \ldots ,n_d ) \in \mathbb{Z}^d ,j = 1, \ldots ,r\}

Nonlinear Frames and Sparse Reconstructions in Banach Spaces

is a Riesz basis of the closed linear spanV0 of