On Orthogonal Polynomials and Finite Moment Problem

Abstract

This paper is an improvement of a previous work on the problem recovering a function or probability density function from a finite number of its geometric moments, [1]. The previous worked solved the problem with the help of the B-Spline theory which is a great approach as long as the resulting linear system is not very large. In this work, two solution algorithms based on the approximate representation of the target probability distribution function via an orthogonal expansion are provided. One primary application of this theory is the reconstruction of the Particle Size Distribution (PSD), occurring in chemical engineering applications. Another application of this theory is the reconstruction of the Radon transform of an image at an unknown angle using the moments of the transform at known angles which leads to the reconstruction of the image form limited data. The aim is to recover a probability density function from a finite number of its geometric moments. The tool is the orthogonal expansion approach. The Shifted-Legendre Polynomials and the Chebyshev Polynomials as bases for the orthogonal expansion are used in this study. A high degree of accuracy has been obtained in recovering a function without facing a possible ill-conditioned linear system, which is the case with many typical approaches of solving the problem. In fact, for a normalized template function f on the interval [0, 1], and a reconstructed function ; the reconstruction accuracy is measured in two domains. One is the moment domain, in which the error (difference between the moments of f and the moments of ) is zero. The other measure is the standard difference in the norm -space ||f- || which can be ≈ 10-6 or less. This paper discusses the problem of recovering a function from a finite number of its geometric moments for the PSD application. Linear transformations were used, as needed, so that the function is supported on the unit interval [0, 1], or on [0, α] for some choice of α. This transformation forces the sequence of moments to vanish. Then, an orthogonal expansion of the Scaled Shifted Legendre Polynomials, as well as the Chebyshev Polynomials, are developed. The result shows good accuracy in recovering different types of synthetic functions. It is believed that up to fifteen moments, this approach is safe and reliable.