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On zeros of entire functions of special form. (English) Zbl 1168.30017
Math. Notes 83, No. 1, 23-30 (2008); translation from Mat. Zametki 83, No. 1, 24-31 (2008).
Let \(\mu\) be a real function of bounded variation on the interval \([0,\sigma]\), \(\sigma> 0\). G. Pólya [Math. Zs. 2, 352–383 (1918; JFM 46.0510.01)] was the first to study the distribution of zeros of entire functions of the form \[ F(z)=\int^\sigma_0 e^{izt}d\mu(t) \] with \(d\mu(t)= g(z)\,dt\), \(g\) integrable, positive, nondecreasing in \((0,\sigma)\). Here, relations beween the functions \(F(z)\) and \[ C(z):=\int^\sigma_0\cos zt\,d\mu(\sigma- t) \] are studied. As a result, sharpening Pólya’s result, the following therorem is proved:
Theorem: Suppose \(C(x)\geq 0\) for all \(x> 0\) and \(F\not\equiv 0\). Then all non-zero real zeros of \(F\) (if they exist) are simple and \(F\) has no zeros in the open lower half-plane \(\text{Im\,}z< 0\).
To prove the first part of the theorem the author shows the relation \[ \text{Re}(F(z) e^{-\sigma z})= -\int^\infty_{-\infty} {y\over\pi(y^2+ (t+ x)^2)}\,C(t)\,dt, \] which under these assumptions is positive for \(\text{Im\,}z< 0\).
To show the simplicity of the zeros the author uses a result of his earlier work on zeros of entire functions [Methods Funct. Anal. Topol. 10, No. 2, 91–104 (2004; Zbl 1054.30024)].
Some examles to apply the theorem are given, for example to entire functions of the form \[ F(z)= \sum^m_{k=0} c_k e^{i\lambda_k z},\quad c_k\in\mathbb{R},\quad 0= \lambda_0< \lambda_1<\cdots< \lambda_m= \sigma. \]

30D15 Special classes of entire functions of one complex variable and growth estimates
30D10 Representations of entire functions of one complex variable by series and integrals
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
30D20 Entire functions of one complex variable, general theory
Full Text: DOI
[1] A. F. Leont’ev, Exponential Series (Nauka, Moscow, 1976) [in Russian].
[2] E. Lukach, Characteristic Functions (Nauka, Moscow, 1979) [in Russian].
[3] A. M. Sedletskii, ”Entire functions of the S. N. Bernstein class that are not Fourier–Stieltjes transforms,” Mat. Zametki 61(3), 367–380 (1997) [Math. Notes 61 (3–4), 301–312 (1997)]. · doi:10.4213/mzm1511
[4] I. V. Tikhonov, ”Uniqueness theorems in linear nonlocal problems for abstract differential equations,” Izv. Ross. Akad. Nauk. Ser. Mat. 67(2), 133–166 (2003) [Izv.Math. 67 (2), 333–363 (2003)]. · Zbl 1073.34071 · doi:10.4213/im429
[5] R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions (Klüwer Academic, Boston-Dordrecht-London, 2004).
[6] G. H. Hardy, ”On the zeroes of certain classes of integral Taylor series. Part II. On the integral function,” Proc. London Math. Soc. Ser. 2 2(876), 401–431 (1905). · JFM 36.0473.03 · doi:10.1112/plms/s2-2.1.401
[7] G. Pólya, ”Über die Nullstellen gewisser ganzer Funktionen,” Math. Z. 2(3–4), 352–383 (1918). · JFM 46.0510.01 · doi:10.1007/BF01199419
[8] E. C. Titchmarsh, ”The zeros of certain integral functions,” Proc. London Math. Soc. Ser. 2 25, 283–302 (1926). · JFM 52.0334.03 · doi:10.1112/plms/s2-25.1.283
[9] M. Cartwright, ”The zeros of certain integral functions,” Quart. J. Math. 1(1), 38–59 (1930). · JFM 56.0973.02 · doi:10.1093/qmath/os-1.1.38
[10] M. Cartwright, ”The zeros of certain integral functions. (II),” Quart. J. Math. 2(1), 113–129 (1931). · JFM 57.0361.03 · doi:10.1093/qmath/os-2.1.113
[11] A.M. Sedletskii, ”On zeros of Laplace transforms of finite measure,” Integral Transform. Spec. Funct. 1(1), 51–59 (1993). · Zbl 0824.44002 · doi:10.1080/10652469308819008
[12] A.M. Sedletskii, ”On the zeros of Laplace transforms,” Mat. Zametki 76(6), 883–892 (2004) [Math. Notes 76 (5–6), 824–833 (2004)]. · Zbl 1079.44001 · doi:10.4213/mzm160
[13] V. P. Zastavnyi, ”A theorem on the zeros of entire functions and its application,” Mat. Zametki 75(2), 192–207 (2004) [Math. Notes 75 (1–2), 175–189 (2004)]. · Zbl 1114.30026 · doi:10.4213/mzm23
[14] V. P. Zastavnyi, ”A theorem on zeros of an entire function and its applications,” Methods Funct. Anal. Topology 10(2), 91–104 (2004). · Zbl 1054.30024
[15] N. I. Akhiezer, Lectures on Integral Transformations (Vishcha Shkola, Kharkov, 1984) [in Russian].
[16] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces (Princeton Univ. Press, Princeton, NJ, 1971; Mir, Moscow, 1974). · Zbl 0232.42007
[17] V. P. Zastavnyi, ”On positive definiteness of some functions,” J. Multivariate Anal. 73(1), 55–81 (2000). · Zbl 0956.42006 · doi:10.1006/jmva.1999.1864
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