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.

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 |