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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 \[ F(z)= \sum^m_{k=0} c_k e^{i\lambda_k z},\quad c_k\in\mathbb{R},\quad 0= \lambda_0< \lambda_1<\cdots< \lambda_m= \sigma. \]

MSC:
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
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References:
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