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Generalization of the Titchmarsh convolution theorem and the complex- valued measures uniquely determined by their restrictions to a half-line. (English) Zbl 0574.30032
Stability problems for stochastic models, Proc. 8th Int. Semin., Uzhgorod/USSR 1984, Lect. Notes Math. 1155, 256-283 (1985).
[For the entire collection see Zbl 0564.00015.]
Let M be the set of all complex valued nonzero Borel measures $$\mu$$ on R such that $$| \mu | (R)$$ is finite. Throughout the following $$j=1,2$$. For $$\mu_ j\in M$$ set $$L(\mu_ j)=\inf \{x:\quad x\in \sup p \mu_ j\}$$ and let $$x_ j=| \mu_ j| (-\infty,x)$$. All the big-oh statements below are as $$x\to -\infty$$. This paper investigates two types of problems.
Part I: Find conditions such that $$L(\mu_ 1*\mu_ 2)=L(\mu_ 1)+L(\mu_ 2)$$. The known conditions are (a) Titchmarsh [cf. B. Ya. Levin, Distribution of zeros of entire functions (1964; Zbl 0152.067)]: $$L(\mu_ j)>-\infty$$. (b) Y. Domar [cf. Proc. Lond. Math. Soc., III. Ser. 46, 288-300 (1983; Zbl 0468.46038)]: there exists $$a>1$$ such that $$x_ j=O(\exp (-| x|^ a))$$. In this paper are proved the weaker conditions: $(c)\quad for\quad all\quad c>0,\quad x_ j=O(\exp (-c| x| \log | x|)).$ (d) For the subclass of M with support on the integers the condition can be weakened to: $\text{for all }c>0,\quad x_ j=O(\exp (-| x| \log | x| -c| x|)).$ These results cannot be improved. (c) and (d) are obtained as corollaries of the following Theorem 2:
Let $$h,g_ j\in H^+$$ and $$h=g_ 1g_ 2$$ and $$g=| g_ 1| +| g_ 2|$$. Further, let $(i)\quad \sup \{g(z):\quad | z| \leq R_ k,\quad z\in C^+\}\leq \exp \exp (o(R_ k))$ for some real sequence $$R_ k\uparrow \infty$$ and (ii) sup$$\{$$ g(z): $$o<Im(z)<\Delta \}<\infty$$ for some $$\Delta >o$$. Then there exist constants $$B_ j$$ such that $$g_ j(z)\exp (iB_ jz)\in H^+$$. The result is not true if little-oh in (i) is repaced by big-oh. However, if $$g_ j$$ are $$2\pi$$ periodic (ii) is automatically satisfied and the bound in (i) can be weakened to exp(o(exp $$R_ k))$$. In this theorem, $$C^+$$ is the half plane $$Im(z)>0$$ and $$H^+$$ is the class of bounded analytic functions in $$C^+.$$
Some of the major steps in the proof of Theorem 2 are given below: 1. Without loss of generality $$g_ j$$ are analytic in $$C^+\cup R$$. 2. B(z) be the Blaschke product corresponding to zeros of $$g_ 1$$. 3. P(z) defined in $$C^+$$ as: $\exp (1/\pi i\int \log | g_ 1(t)| \{1/(t-z)-t/(1+t^ 2)\}dt)$ is in $$H^+$$. 4. G(z) defined as i log$$\{g_ 1(z)/B(z)P(z))\}$$ for z in $$C^+$$ can be extended to an entire function (also called G(z)). $(5)\quad | G(z)| \leq \exp (o(R_ k))\quad for\quad | z| =R_ k\uparrow \infty.$ $(6)\quad \sup_{| s| <\Delta}\int (| Im G(t+is)| /(1+t^ 2))dt<\infty.$ (7) $$| G(z)| =o(| z|^ 3)$$ as $$| z| \to \infty$$, $$| Im z| \leq \Delta /2$$. 8. G(z) is linear in z with real coefficients.
Part II: Let E and F be subclasses of measures $$\mu_ j\in M$$ that satisfy the conditions (e) and (f), respectively. Here (e) $$L(\mu_ j)=- \infty$$ and $for\quad all\quad c>0,\quad x_ j=O(\exp (-c| x|)),$ (f) $$L(\mu_ j)=-\infty$$ and $for\quad all\quad c>0,\quad x_ j=O(\exp (-c| x| \log | x|)).$ Note that $$F\subset E$$. Let $$E_ n=\{\mu^{n^*}:$$ $$\mu\in E\}$$ with $$F_ n$$ similarly defined. The following results are proved: Let $$n\geq 3$$ and $$v_ j\in E_ n$$ (or $$v_ j\in F_ n)$$. If, for some real a, the restrictions of $$v_ 1$$ and $$v_ 2$$ to the half line (-$$\infty,a)$$ coincide then $$v_ 1\equiv v_ 2$$. These results are not true for $$n=2$$. Several known results are shown to be special cases of these results. The proofs are too involved to be summarized here.
Reviewer: R.Shantaram

##### MSC:
 30D15 Special classes of entire functions of one complex variable and growth estimates 30E99 Miscellaneous topics of analysis in the complex plane 62E10 Characterization and structure theory of statistical distributions