Around operators not increasing the degree of polynomials

TL;DR

This paper investigates linear operators on polynomial spaces that do not increase polynomial degree, characterizing polynomial sequences invariant under such operators, with applications to operators with limited-term expansions.

Contribution

It introduces a general framework for analyzing degree-preserving linear operators on polynomials and characterizes polynomial sequences invariant under these operators.

Findings

Characterization of polynomial sequences invariant under degree-preserving operators

Development of a dual space approach for the problem

Application to operators with three-term expansions

Abstract

We present a generic operator $J$ simply defined as a linear map not increasing the degree from the vectorial space of polynomial functions into itself and we address the problem of finding the polynomial sequences that coincide with the (normalized) $J$-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. It is also provided examples where the results are applied to the case where $J$'s expansion is limited to three terms.