This paper presents a new approach for tackling the shift-invariance problem in the discrete Haar domain, without trading off any of its desirable properties, such as compression, separability, orthogonality, and symmetry. The paper presents several key theoretical contributions. First, we derive closed form expressions for phase shifting in the Haar domain both in partially decimated and fully...

the aim of this paper is a) if σak2 < ∞ then σak rk(x) is in bmo that{rk(x)} is rademacher system. b) p1k=1 ak!nk (x) 2 bmo; 2k  nk < 2k+1that f!n(x)g is walsh system. c) if jakj < 1k then p1k=1 ak!k(x) 2 bmo.

In order to get an efficient image representation we introduce a new adaptive Haar wavelet transform, called Tetrolet Transform. Tetrolets are Haar-type wavelets whose supports are tetrominoes which are shapes made by connecting four equal-sized squares. The corresponding filter bank algorithm is simple but enormously effective. In every level of the filter bank algorithm we divide the low-pass...

Long term memory has frequently been observed in physical time series. Statistical theory for long term memory stochastic processes is radically different from the standard time series analysis, which assumes short term memory. The Allen variance is a particular measure of variability developed for long term memory processes. This variance can be interpreted as a Haar wavelet coefficient varian...

ABSTRACT: This paper begins by using the Haar wavelet to analyze the cumulative distribution function of a stream of two-valued Bernoulli trials. We find that this function maps binary numbers in [0,1] into non-uniform binary numbers in [0,1]. More generally, for r-valued Bernoulli trials, the distribution function likewise maps usual base-r numbers into non-uniform base-r numbers. We also find...

A new computational method based on Wilson wavelets is proposed for solving a class of nonlinear stochastic It^{o}-Volterra integral equations. To do this a new stochastic operational matrix of It^{o} integration for Wilson wavelets is obtained. Block pulse functions (BPFs) and collocation method are used to generate a process to forming this matrix. Using these basis functions and their operat...

In order to get an efficient image representation we introduce a new adaptive Haar wavelet transform, called Tetrolet Transform. Tetrolets are Haar-type wavelets whose supports are tetrominoes which are shapes made by connecting four equal-sized squares. The corresponding filter bank algorithm is simple but enormously effective. In every level of the filter bank algorithm we divide the low-pass...