The Beurling theorem for continuous limited functions and functions of Stepanov with discrete spectrum. (Russian. English summary) Zbl 1334.46021

Kaluzhina, N. S.; Marushenkov, S. V.

Vestn. Voronezh. Gos. Univ., Ser. Fiz. Mat. 2008, No. 2, 115-121 (2008).

Let \(\psi_n\) (\(n=1,2,\dots\)) and \(\psi\) be bounded continuous functions on \(\mathbb R\). We say that \(\psi_n\to \psi\) narrowly as \(n\to\infty\) if \(\|\psi_n\|_\infty \to \|\psi\|_\infty\) and \(\psi_n(x)\to \psi(x)\) uniformly on finite intervals for \(n\to\infty\). The classical A. Beurling theorem [Acta Math. 77, 127–136 (1945; Zbl 0061.13311)] states: Let \(f\not\equiv 0\) bounded and uniformly continuous on \(\mathbb R\), then there are a number \(\lambda\in\mathbb R\) and a sequence \(\{\psi_n\}\) of linear combinations of translates of \(f\) such that \(\psi_n(x)\to \exp(i\lambda x)\) narrowly as \(n\to\infty\). In the paper under review, the authors prove some analogues of the Beurling theorem for some bounded continuous functions and some functions from the Stepanov space.

Reviewer: Sergei S. Platonov (Petrozavodsk)

MSC:

46E15	Banach spaces of continuous, differentiable or analytic functions
28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence

Keywords:

Beurling spectrum; Beurling theorem; Stepanov space; space of continuous bounded functions

Citations:

Zbl 0061.13311