On the closure of translation-dilation invariant linear spaces of polynomials

TL;DR

This paper proves that the pointwise limit of a converging sequence of polynomials in a translation and dilation invariant linear space remains within that space, establishing a closure property.

Contribution

It establishes the closure of translation-dilation invariant polynomial spaces under pointwise limits, a new theoretical result in polynomial space analysis.

Findings

Pointwise limits of sequences in the space stay within the space.

Closure property holds for translation-dilation invariant polynomial spaces.

The result applies to real polynomials in multiple variables.

Abstract

Assume that a linear space of real polynomials in $d$ variables is given which is translation and dilation invariant. We show that if a sequence in this space converges pointwise to a polynomial, then the limit polynomial belongs to the space, too.