Weighted versions of Beurling's Tauberian theorem (Q1070465)

Beurling's version of Wiener's Tauberian theorem [as published by \textit{T. T. Moh}, Proc. Am. Math. Soc. 36, 162-172 (1972; Zbl 0254.40014) and \textit{G. E. Peterson}, J. Lond. Math. Soc., II. Ser. 5, 182-190 (1972; Zbl 0236.40010) respectively] states, that for any given self-neglecting function s(x), such as typically \(s(x)=(1+| x|)^{\alpha}\) for \(\alpha\in [0,1)\), one has: If for \(f\in L^{\infty}({\mathbb{R}})\), \(g_ 0\in L^ 1({\mathbb{R}})\) the relation \(\lim_{x\to \infty}s^{-1}(x)\int_{{\mathbb{R}}}f(y)g_ 0((x-y)| s(x))dy=0\) holds, the same relation holds true for any other \(g\in L^ 1({\mathbb{R}})\) if (and only if) \(\hat g_ 0(t)\neq 0\) for any \(t\in {\mathbb{R}}\). In the present note weighted \((=quantitative)\) versions of this result are given, for s as above, and appropriate speed of convergence to zero for \(x\to \infty\). Since the moving-average procedure involved in the argument does not commute with translation nor defines an associate (convolution) product a more refined analysis and delicate estimates are required to push through the general approach due to N. Wiener in this setting.