Ultradifferentiable functions via the Laguerre operator

TL;DR

This paper introduces a new way to define and analyze ultradifferentiable functions on positive real space using the Laguerre operator, linking them to Pilipovi spaces through series expansion properties.

Contribution

It provides a novel characterization of ultradifferentiable functions via Laguerre series and establishes an isomorphism with Pilipovi spaces, expanding the functional analysis framework.

Findings

Characterization of ultradifferentiable functions using Laguerre series coefficients

Establishment of an isomorphism between ultradifferentiable function spaces and Pilipovi spaces

Extension of the theory of ultradifferentiable functions on  spaces

Abstract

We define and characterize ultradifferentiable functions and their corresponding ultradistributions on $\RR^d_+$ using iterates of the Laguerre operator. The characterization is based on decay or growth conditions of the coefficients in their Laguerre series expansion. We apply our results to establish an isomorphism between subspaces of Pilipovi\'c spaces on $\RR^d$, and the spaces of ultradifferentiable functions on $\RR^d_+$.