The Fitting of Power Series, Meaning Polynomials, Illustrated on Band-Spectroscopic Data

TLDR

The prototype of fitting polynomials to equally‑spaced data arises in band‑spectra analysis, and similar considerations apply when multiple variables or nonlinear fitting problems are involved. The study reexamines polynomial fitting by addressing problem formulation, robust regression techniques, coordinate selection, least‑squares properties, stopping criteria, and result presentation. The findings show that the insights gained extend to other contexts involving sums of single‑variable functions.

Abstract

The prototype of fitting polynomials to equally-spaced data—in which the equalspacing is theoretically precise and the data is accurate to many decimal places—arises in the analysis of band spectra. A hard look at such examples forces us to reexamine our thinking on such diverse issues as: How to formulate such problems, the use of robust/resistant techniques in polynomial regression, which coordinates to use and why, the basic properties of linear least squares, choices in stopping a fit, and improved ways to describe our answers. Our results and attitudes apply rather directly to other situations where we are fitting a sum of functions of a single variable. When two or more different variables, subject to error, blunder, or omission, underlie the carriers to be considered, regression/fitting problems are likely to need not only the considerations presented here, but others as well. To a varying extent, the same will be true of nonlinear fitting/regression problems.