مجله موجکها و جبر خطی
سال چهارم شماره 2 (Autumn and Winter 2017)

In ýthe present ýpaper, ýwe ýintroduce the ýtwo-wavelet ýlocalization ýoperator ýfor ýthe square ýintegrable ýrepresentation ýof aý ýhomogeneous spaceý with respect to a relatively invariant measure. ýWe show that it is a bounded linear operator. We investigate ýsome ýproperties ýof the ýtwo-wavelet ýlocalization ýoperator ýand ýshow ýthat ýit ýis aý ýcompact ýoperator ýand is ýcontained ýiný a Schatten $p$-classý.

In this paper, we first give a characterization of sub-topical functions with respect to their lower level sets and epigraph. Next, by using two different classes of elementary functions, we present a characterization of sub-topical functions with respect to their polar functions, and investigate the relation between polar functions and support sets of this class of functions. Finally, we obtain more results on the polar of sub-topical functions.

Miranda-Thompson majorization is a group-induced cone ordering on Rn induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize Miranda-Thompson majorization on the matrices. For X, Y 2 Mm;n, X is said to be Miranda- Thompson majorized by Y (denoted by X mt Y ) if there exists some D 2 Conv(G) such that X = DY . Also, we characterize linear preservers of this concept on Mm;n.

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 operational matrices of integration and stochastic integration, the problem under study is transformed to a system of nonlinear algebraic equations which can be simply solved to obtain an approximate solution for the main problem. Moreover, a new technique for computing nonlinear terms in such problems is presented. Furthermore, convergence of Wilson wavelets expansion is investigated. Several examples are presented to show the eciency and accuracy of the proposed method.

Can we characterize the wavelets through linear transformation? the answer for this question is certainly YES. In this paper we have characterized the Haar wavelet matrix by their linear transformation and proved some theorems on properties of Haar wavelet matrix such as Trace, eigenvalue and eigenvector and diagonalization ofa matrix.

This paper introduces an inequality on vectors in $\mathbb{R}^n$ which compares vectors in $\mathbb{R}^n$ based on the $p$-norm of their projections on $\mathbb{R}^k$ ($k\leq n$).
For $p>0$, we say $x$ is $d$-projectionally less than or equal to $y$ with respect to $p$-norm if $\sum_{i=1}^k\vert x_i\vert^p$ is less than or equal to $ \sum_{i=1}^k\vert y_i\vert^p$, for every $d\leq k\leq n$. For a relation $\sim$ on a set $X$, we say a map $f:X \rightarrow X$ is a preserver of that relation, if $x\sim y$ implies $f(x)\sim f(y)$, for every $x,y\in X$. All the linear maps that preserve $d$-projectional equality and inequality are characterized in this paper.