Uncertainty Principles and Balian-Low type Theorems in Principal Shift-Invariant Spaces

In this paper, we consider the time-frequency localization of the generator of a principal shift-invariant space on the real line which has additional shift-invariance. We prove that if a principal shift-invariant space on the real line is translation-invariant then any of its orthonormal (or Riesz) generators is non-integrable. However, for any $n\ge2$, there exist principal shift-invariant spaces on the real line that are also \nZ-invariant with an integrable orthonormal (or a Riesz) generator $\phi$, but $\phi$ satisfies $\int_{\mathbb R} |\phi(x)|^2 |x|^{1+\epsilon} dx=\infty$ for any $\epsilon>0$ and its Fourier transform $\hat\phi$ cannot decay as fast as $(1+|\xi|)^{-r}$ for any $r>1/2$. Examples are constructed to demonstrate that the above decay properties for the orthormal generator in the time domain and in the frequency domain are optimal

Oversampling of wavelet frames for real dilations

We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace.Comment: Journal of London Mathematical Society, published online March 13, 2012 (to appear in print

The near shift-invariance of the dual-tree complex wavelet transform revisited

The dual-tree complex wavelet transform (DTCWT) is an enhancement of the conventional discrete wavelet transform (DWT) due to a higher degree of shift-invariance and a greater directional selectivity, finding its applications in signal and image processing. This paper presents a quantitative proof of the superiority of the DTCWT over the DWT in case of modulated wavelets.Comment: 15 page

Shannon Multiresolution Analysis on the Heisenberg Group

We present a notion of frame multiresolution analysis on the Heisenberg group, abbreviated by FMRA, and study its properties. Using the irreducible representations of this group, we shall define a sinc-type function which is our starting point for obtaining the scaling function. Further, we shall give a concrete example of a wavelet FMRA on the Heisenberg group which is analogous to the Shannon MRA on \RR.Comment: 17 page