On the frequency function of periodic quasianalytic and non-quasianalytic functions
Abstract
For a real-valued continuous periodic function f, let Ar f(x)=1/2r∫x−rx+rf(t) dt, A0f(x)=f(x), and let rf(x) be the smallest radius at which supr≥ 0Ar f(x) is attained. We call rf the frequency function of f. Steinerberger proved that a two-valued condition involving both rf and r−f characterizes a single sinusoidal function, and asked if the same result holds under the assumption that rf has finite range (or even range with Lebesgue measure zero). In this paper, we prove that a stronger finite-valued condition for rf characterizes a sinusoidal function for the case when f is periodic, smooth, and belongs to a quasianalytic Denjoy-Carleman class. However, we also show that the conjecture of Steinerberger is not true in general, since we can construct non-sinusoidal periodic smooth functions in non-quasianalytic Denjoy-Carleman classes such that their frequency functions have finite range.
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BibTeX
@misc{garzon2026frequency,
title = {{On the frequency function of periodic quasianalytic and non-quasianalytic functions}},
author = {Carlos Garzón and José Madrid},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24953},
eprint = {2610.24953},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24953},
}
RIS
TY - UNPB
TI - On the frequency function of periodic quasianalytic and non-quasianalytic functions
AU - Carlos Garzón
AU - José Madrid
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.24953
UR - https://arxiv.org/abs/2610.24953
AB - For a real-valued continuous periodic function f, let A_r f(x)=1/2r∫_x−r^x+rf(t) dt, A_0f(x)=f(x), and let r_f(x) be the smallest radius at which sup_r≥ 0A_r f(x) is attained. We call r_f the frequency function of f. Steinerberger proved that a two-valued condition involving both r_f and r_−f characterizes a single sinusoidal function, and asked if the same result holds under the assumption that r_f has finite range (or even range with Lebesgue measure zero). In this paper, we prove that a stronger finite-valued condition for r_f characterizes a sinusoidal function for the case when f is periodic, smooth, and belongs to a quasianalytic Denjoy-Carleman class. However, we also show that the conjecture of Steinerberger is not true in general, since we can construct non-sinusoidal periodic smooth functions in non-quasianalytic Denjoy-Carleman classes such that their frequency functions have finite range.
ER -