On non-quasianalytic classes of infinitely differentiable functions

G.S. Balashova

National Research University “Moscow Power Engineering Institute”, Moscow, 111250 Russia

E-mail: balashovags@mpei.ru

Received August 31, 2021

ORIGINAL ARTICLE

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DOI: 10.26907/2541-7746.2022.1.43-59

For citation: Balashova G.S. On non-quasianalytic classes of infinitely differentiable functions. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2022, vol. 164, no. 1, pp. 43–59. doi: 10.26907/2541-7746.2022.1.43-59. (In Russian)

Abstract

This article investigates the connection between two positive logarithmically convex sequences {M̂_n} and {M_n}, which define respectively the Carleman classes of functions infinitely differentiable on the set J and sequences {b_n} 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 L_r (J), not only for the classical case r = ∞ but also for any r ≥ 1. Obviously, M_n ≤ M̂_n  is always observed. Here the sequences {M̂_n}, 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

References

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