Approximation in a class of Banach algebras of quasianalytically smooth analytic functions
The purpose of this important paper in harmonic analysis is to prove a Nyman-Gurarii-Levin type approximation theorem going far beyond the classical theorems of Wiener for \(L^ 1(\mathbb{R})\) and those of Beurling for weighted \(L^ 1\) spaces on \(\mathbb{R}\). Let us recall that Wiener's theorem states that the linear space of all (right and left) translates of a collection of functions in \(L^ 1(\mathbb{R})\) is norm dense in \(L^ 1(\mathbb{R})\) if and only if the Fourier transforms of the functions in the given set have no common zeros on \(\mathbb{R}\). Beurling generalized this result to weighted \(L^ 1\) spaces on \(\mathbb{R}\) for (submultiplicative) weights \(\omega\) of nonquasianalytic type, i.e., which satisfy \(\int^ \infty_{-\infty} (1+ t^ 2)^{- 1}\log \omega(t)dt< \infty\). Later it was shown by \textit{A. Vretblad} [Ark. Mat. 11, 109-138 (1973; Zbl 0258.46047)] and \textit{Y. Domar} [Lect. Notes Math. 975, 210-213 (1983; Zbl 0501.47009)] that the convergence of this integral is indeed necessary in order to have a Wiener type analogue for \(L^ 1(\mathbb{R},\omega)\). Extensions of the Wiener-Beurling theorem to \(L^ 1(\mathbb{R}_ +)\) and \(L^ 1(\mathbb{R}_ +,\omega)\) were given by \textit{Nyman} and \textit{Gurarii- Levin} [Zap. Meh. Mat. Fak. i Harkov. Mat. Obshch. 30, 178-185 (1964)], respectively. Their methods made it necessary to work with nonquasianalytic weights. The examples, however, which show that Beurling's theorem does not extend to quasianalytic weights on \(\mathbb{R}\), do not carry over to the half line situation. Gurarii now conjectures that Nyman's theorem should extend to spaces \(L^ 1(\mathbb{R}_ +,\omega)\), no matter whether \(\omega\) is quasianalytic or not. In the present paper, the authors prove a result on the denseness of right translates for a class of Banach algebras \(Q(\mathbb{C}_ -,w)\) of analytic functions on the open lower half-plane \(\mathbb{C}_ -\) which strongly supports Gurarii's conjecture. In order to treat both the quasianalytic and nonquasianalytic case, new powerful methods had to be developed; a milestone in the theory of harmonic analysis. \{Reviewer's remark: Meanwhile the authors could give a positive answer to Gurarii's conjecture in their paper: Completeness of translates in weighted Hilbert spaces on the half-plane, to appear in Acta Math.\}.
Nyman-Gurarii-Levin type approximation theorem
scientific article; zbMATH DE number 515234