Stability of third degree linear functionals and rational spectral transformations. (English) Zbl 1492.42022
Summary: In this paper, we consider linear forms of the third degree class. This means that the Stieltjes function associated with the corresponding moment sequence satisfies a cubic equation with polynomial coefficients. We introduce the notion of a primitive triple of a strict third degree form. A simplification criterion of the corresponding cubic algebraic equation is given. Moreover, we show that the class of third degree linear forms is closed under rational spectral transformations. Several consequences of this fact are deduced. In particular, we illustrate with several examples the set of third degree linear forms is stable for the most standard algebraic operations in the linear space of linear forms.
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Keywords:
orthogonal polynomials; Stieltjes function; third degree forms; rational spectral transformations; associated forms; perturbed formsReferences:
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