### Journals Information

**
Mathematics and Statistics Vol. 10(3), pp. 653 - 658 DOI: 10.13189/ms.2022.100322 Reprint (PDF) (306Kb) **

## A Bounded Maximal Function Operator and Its Acting on Functions

**Raghad S.Shamsah ^{*}**

Department of Petroleum Eng., Engineering Faculty, Kerbala University, Iraq

**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 functions with respect to 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 and under the assumption that scaling wavelet function of a given multiresolution analysis is spherical wavelet with 0-regularity, the convergence of ) 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 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 . 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 is regarded to be a key in solving convergence problems that occur inside spaces dimension .

**KEYWORDS**

Spherical Scaling Spherical Multiresolution Analysis, Wavelet Expansions, Convergence, Rapidly Decreasing, Maximal Function

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Raghad S.Shamsah , "A Bounded Maximal Function Operator and Its Acting on Functions," Mathematics and Statistics, Vol. 10, No. 3, pp. 653 - 658, 2022. DOI: 10.13189/ms.2022.100322.

(b). APA Format:

Raghad S.Shamsah (2022). A Bounded Maximal Function Operator and Its Acting on Functions. Mathematics and Statistics, 10(3), 653 - 658. DOI: 10.13189/ms.2022.100322.