Banach algebras of measures on the real line with a given asymptotics of distributions at infinity. (English. Russian original) Zbl 0936.46021

Rogozin, B. A.; Sgibnev, M. S.

Sib. Math. J. 40, No. 3, 565-576 (1999); translation from Sib. Mat. Zh. 40, No. 3, 660-672 (1999).

Let \(\nu\) be a \(\sigma\)-additive complex-valued measure defined on the bounded Borel subsets of \({\mathbb R}\). Let \(\mu^{n*}\) be the \(n\)th convolution of \(\mu\). Given \(\gamma>0\), the set \(S(\gamma)=\{\mu':\int_{\mathbb R}e^{\gamma x}|\mu'|(dx)<\infty\}\) is a Banach algebra of measures. The authors assume that \(\mu\in S(\gamma)\). Let \(G\in S(\gamma)\) be an unbounded probability distribution on \({\mathbb R}^+\) such that
(i) the limit \(\lim_{x\to\infty}G^{2*}([x,\infty))/G([x,\infty))= 2\widehat\mu(\gamma)\) exists, where \(\widehat\mu(\gamma)=\int_{\mathbb R}e^{\gamma x}\mu(dx)\);
(ii) for every fixed \(y\), \(G([x+y,\infty))/G([x,\infty))\to e^{-\gamma y}\) as \(x\to\infty\).
Let \(f(z)=\sum_{n=0}^\infty a_n z^n\) be an analytic function in some domain \(D\subset{\mathbb C}\) entirely including the spectrum of \(\mu\). If the limit of the relation \(\mu([x,\infty))/G([x,\infty))\) as \(x\to\infty\) exists then the measure \(f(\mu)=\sum_{n=0}^\infty a^n\mu^{n*}\) has the following property: \[ f(\mu)([x,\infty)) \sim f'(\widehat\mu(\gamma))\mu([x,\infty)), \quad x\to \infty. \]
The article is a continuation of the paper by the second author [Sib. Math. J. 29, No. 4, 647-655 (1988; Zbl 0667.46016)].

Reviewer: D.A.Korshunov (Novosibirsk)

MSC:

46E27	Spaces of measures
46H05	General theory of topological algebras
60E99	Distribution theory

Keywords:

Banach algebra of measures; convolution; unbounded probability distribution

Citations:

Zbl 0667.46016

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