On non-quasianalytic classes of infinitely differentiable functions

Abstract

This article investigates the connection between two positive logarithmically convex sequences {M̂n} and {Mn}, which define respectively the Carleman classes of functions infinitely differentiable on the set J and sequences {bn} specifying the values of the function itself and all its derivatives at some point x0 ∈ J. The results obtained are more general than those previously known, and explicit constructions of the required functions are proposed with estimates for the norms of the functions themselves and their n derivatives in the Lebesgue spaces Lr (J), not only for the classical case r = ∞ but also for any r ≥ 1. Obviously, Mn ≤ M̂n is always observed. Here the sequences {M̂n}, for which equality holds, are indicated and specific examples are given.

Keywords

• non-quasianalytical carleman classes
• logarithmically convex
• condition sequence
• existence
• function
• satisfying
• construction
• regularization
• fundamental indices