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

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)].

### MSC:

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

Zbl 0667.46016
Full Text:

### References:

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