On invertibility conditions for elements of Banach algebras of measures (Q358177)

Let \(S(\phi)\) be a certain commutative Banach algebra of measures on \(\mathbb{R}\) with the convolution of measures as the multiplication. Let \(\widehat{\nu}\) denote the Laplace transform of the measure \(\nu \in S(\phi) \). It is proved that, if \(\nu = \nu_c + \nu_d + \nu_s \) is the decomposition of an element \(\nu \in S(\phi)\) into an absolutely continuous, discrete, and singular component, respectively, then the element \(\nu\) has an inverse if \(\widehat{\nu}(s) \neq 0 \) for every \(s\in \Pi:=\{z: \omega_0 \leq \mathrm {Re}\,\omega_1\}\) and if \[ \inf_{s \in \Pi} | \widehat{\nu_d}(s)| \geq \max_{i=0,1} \int_{\mathbb{R}} e^{\omega_i x}|\nu_s|(dx) . \]