Abstract
We characterize the equality between ultradifferentiable function classes defined in terms of abstractly given weight matrices and in terms of the corresponding matrix of associated weight functions by using new growth indices. These indices, defined by means of weight sequences and (associated) weight functions, are extending the notion of O-regular variation to a mixed setting. Hence we are extending the known comparison results concerning classes defined in terms of a single weight sequence and of a single weight function and give also these statements an interpretation expressed in O-regular variation.
Highlights
In the theory of ultradifferentiable function spaces there exist two classical approaches in order to control the growth of the derivatives of the functions belonging to such classes: Either one uses a weight sequence M = (Mj)j or a weight function ω : [0, +∞) → [0, +∞)
Motivated by the results from [1], in [14] and [11] ultradifferentiable classes defined in terms of weight matrices M have been introduced and it has been shown that, by using the weight matrix Ω associated with a given weight function ω, in this general framework one is able to treat both classical settings in a uniform and convenient way and more classes
Since to each weight function ω we can associate a weight matrix Ω and since it is known that to each sequence M one can associate a weight function ωM, in [14, Sect. 9.3] and in [15] the following iterated process has been studied: When I = R>0 is denoting the index set, by starting with an abstractly given matrix M := {M x : x ∈ I} with some regularity properties we immediately get the weight function matrix ωM := {ωMx : x ∈ I}
Summary
In the theory of ultradifferentiable function spaces there exist two classical approaches in order to control the growth of the derivatives of the functions belonging to such classes: Either one uses a weight sequence M = (Mj)j or a weight function ω : [0, +∞) → [0, +∞). In both settings one requires several basic growth and regularity assumptions on M and ω and one distinguishes between two types, the Roumieu type spaces E{M} and E{ω}, and the Beurling type spaces E(M) and E(ω). In the following we write E[ ] for all arising classes
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