A Bounded Maximal Function Operator and Its Acting on <img src=image/13425593_01.gif> Functions

Abstract

With a novel generation operator known as Spherical Scaling Wavelet Projection Operator, this study proposes new strategies for achieving the scaling wavelet expansion's convergence of <img src=image/13425593_01.gif> functions with respect to <img src=image/13425593_02.gif> almost everywhere under generic hypotheses. Hypotheses of results are based on three types of conditions: Space's function f, Kind of Wavelet functions (spherical) and Wavelet Conditions. The results showed that in the case of <img src=image/13425593_03.gif> and under the assumption that scaling wavelet function <img src=image/13425593_04.gif> of a given multiresolution analysis is spherical wavelet with 0-regularity, the convergence of <img src=image/13425593_01.gif>) expansions almost everywhere will be achieved under a new kind of partial sums operator. We can examine some properties of spherical scaling wavelet functions like rapidity of decreasing and boundedness. After estimating the bounds of spherical scaling wavelet expansions, we examined the limited (bounds) of this operator. The results are established on the almost everywhere wavelet expansions convergence of <img src=image/13425593_01.gif> space functions. Several techniques were followed to achieve this convergence, such as the bounded condition of the Spherical Hardy-Littlewood maximal operator is achieved using the maximal inequality and Riesz basis functions conditions. The general wavelet expansions' convergence was demonstrated using the spherical scaling wavelet function and several of its fundamental features. In fact, the partial sums in these expansions are dominated in their magnitude by the maximal function operator, which may be applied to establish convergence. The convergence here may be obtained by assuming minimal regularity for a spherical scaling wavelet function <img src=image/13425593_05.gif>. The focus of this research is on recent advances in convergence theory issues with respect to spherical wavelet expansions' partial sums operators. The employment of scaling wavelet basis functions defined on <img src=image/13425593_06.gif> is regarded to be a key in solving convergence problems that occur inside spaces dimension <img src=image/13425593_06.gif>.