Generalization of the Titchmarsh convolution theorem and the complex- valued measures uniquely determined by their restrictions to a half-line. (English) Zbl 0574.30032

Ostrovskii, I. V.

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

Keywords:

complex valued measures; Titchmarsh convolution theorem; Borel measures; bounded analytic functions; Blaschke product