# Review and Improvement of the Finite Moment Problem

Fawaz Hjouj1, *, Mohamed Soufiane Jouini1
1 Deparment of Mathematics, Khalifa University, Abu Dhabi, UAE

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: https://creativecommons.org/licenses/by/4.0/legalcode. This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Deparment of Mathematics, Khalifa University, UAE Box 2533 Abu Dhabi, UAE; Tel: +971 2 6075186; E-mail: fawaz.hjouj@ku.ac.ae

## Abstract

### Background:

This paper reviews the Particle Size Distribution (PSD) problem in detail. Mathematically, the problem faced while recovering a function from a finite number of its geometric moments has been discussed with the help of the Spline Theory. Undoubtedly, the splines play a major role in the theory of interpolation and approximation in many fields of pure and applied sciences. B-Splines form a practical basis for the piecewise polynomials of the desired degree. A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula has been tested on several types of synthetic functions. This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basis functions and the reduction of the size along with an appropriate transformation of the resulting linear system for stability.

### Objective:

The aim is to recover a function from a finite number of its geometric moments.

### Methods:

The main tool is the Spline Theory. Undoubtedly, the role of splines in the theory of interpolation and approximation in many fields of pure and applied sciences has been well- established. B-Splines form a practical basis for the piecewise polynomials of the desired degree.

### Results:

A high degree of accuracy has been obtained in recovering a function within the first ten to fifteen geometric moments. The proposed approximation formula is tested on several types of synthetic functions.

### Conclusion:

This work highlights some advantages, such as the use of a practical basis for the approximating space, the exactness of computing the moments of these basic functions and the reduction of the size along with the data transformation of the resulting linear system for stability.

Keywords: Reconstruction of a function, Moments, Splines, Orthogonal Polynomials, Particle Size Distribution, PSD reconstruction, Kernel Density Function-Based Method (KDF).