On properties of a lattice structure for a Wavelet Filter Bank Implementation

This paper presents concept of a lattice structure for parametrization and implementation of a Discrete Wavelet Transform. Theoretical properties of the lattice structure are discussed in detail. An algorithm for converting the lattice structure to a wavelet filter bank coeffcients is constructed. A theoretical proof demonstrating that filters implemented by the lattice structure fulfil conditions imposed on an orthogonal wavelet filter bank is conducted

On properties of a lattice structure for a wavelet filter bank implementation. Pt. 2

This paper continues discussion of a lattice structure for parametrization and implementation of a Discrete Wavelet Transform. Based on an algorithm for converting the lattice structure to a wavelet filter bank coefficients, developed in the first part of this paper, second part of the proof demonstrating that filters implemented by the lattice structure fulfil conditions imposed on an orthogonal wavelet filter bank is carried out

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

Multiresolution analysis in statistical mechanics. I. Using wavelets to calculate thermodynamic properties

The wavelet transform, a family of orthonormal bases, is introduced as a technique for performing multiresolution analysis in statistical mechanics. The wavelet transform is a hierarchical technique designed to separate data sets into sets representing local averages and local differences. Although one-to-one transformations of data sets are possible, the advantage of the wavelet transform is as an approximation scheme for the efficient calculation of thermodynamic and ensemble properties. Even under the most drastic of approximations, the resulting errors in the values obtained for average absolute magnetization, free energy, and heat capacity are on the order of 10%, with a corresponding computational efficiency gain of two orders of magnitude for a system such as a $4\times 4$ Ising lattice. In addition, the errors in the results tend toward zero in the neighborhood of fixed points, as determined by renormalization group theory.Comment: 13 pages plus 7 figures (PNG

Tensor network and (pp-adic) AdS/CFT

We use the tensor network living on the Bruhat-Tits tree to give a concrete realization of the recently proposed $p$-adic AdS/CFT correspondence (a holographic duality based on the $p$-adic number field $\mathbb{Q}_p$). Instead of assuming the $p$-adic AdS/CFT correspondence, we show how important features of AdS/CFT such as the bulk operator reconstruction and the holographic computation of boundary correlators are automatically implemented in this tensor network.Comment: 59 pages, 18 figures; v3: improved presentation, added figures and reference