On generalized definitions of ultradifferentiable classes

TL;DR

This paper unifies various ultradifferentiable function classes under a general framework using weight matrices, extending known results to new classes defined beyond traditional growth factors.

Contribution

It demonstrates that certain ultradifferentiable classes are special cases of a broader framework based on weight matrices, and extends results to classes beyond geometric growth.

Findings

Ultradifferentiable classes can be represented by weight matrices.

New classes defined by weight and exponent sequences are compatible with existing frameworks.

Results from matrix-type classes transfer to these new non-standard classes.

Abstract

We show that the ultradifferentiable-like classes of smooth functions introduced and studied by S. Pilipovi\'c, N. Teofanov and F. Tomi\'c are special cases of the general framework of spaces of ultradifferentiable functions defined in terms of weight matrices in the sense of A. Rainer and the third author. We study classes "beyond geometric growth factors" defined in terms of a weight sequence and an exponent sequence, prove that these new types admit a weight matrix representation and transfer known results from the matrix-type to such a non-standard ultradifferentiable setting.