TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 4, April 2009, Pages 2207–2223 S 0002-9947(08)04714-4 Article electronically published on November 17, 2008 ON ZEROS OF SOME ENTIRE FUNCTIONS ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK Abstract. We study the distribution of zeros z k for entire functions of the form sin z +  1 0 f (t)e iz(1−2t) dt with f belonging to a space X → L 1 (0, 1). For a large class X of spaces X (including, e.g., the spaces L p (0, 1) for all p ∈ [1, ∞]) we show that z k = πk +ζ k , where (ζ k ) k∈Z is the sequence of Fourier coefficients for some function g in X, and study properties of the induced mapping g → f . 1. Introduction The aim of this paper is to study the distribution of zeros for a class of entire functions of the form (1.1) F (z)= F f (z) := sin z +  1 0 f (t)e iz(1−2t) dt, where f is a function integrable over (0, 1). Such a class consists of functions of the so-called sine type [12, Lect. 22]. Moreover, to this form also reduce the functions ˜ F (z)= m − e −iz + m + e iz +  1 −1 ˜ f (t)e izt dt, m − m + =0, which are the Fourier–Stieltjes transforms [7, Ch. 12.5] of the measures m − δ −1 + m + δ 1 + ˜ f (t) dt on [−1, 1], with δ x being the Dirac measure at the point x and ˜ f ∈ L 1 (−1, 1). Indeed, taking α := π 2 + 1 2i log(m − /m + ) and c := −2 √ m − m + , we get by direct calculations the equality ˜ F (z + α)= cF f (z) for a suitable f ∈ L 1 (0, 1). Asymptotic distribution of zeros for Fourier–Stieltjes transforms is of key sig- nificance in many areas of function theory, harmonic analysis, functional analysis, etc. and has been studied for many particular situations; see, e.g., [10, 11, 19]. Our interest in this topic stems from the spectral theory of Sturm–Liouville and Dirac operators, since their characteristic functions usually are of the above type; e.g., the case f ∈ W 1 1 (0, 1) appears in [15], f ∈ W 1 2 (0, 1) in [16, 17], f ∈ W s 2 (0, 1) with s ∈ [0, 1] in [9], f ∈ L p (0, 1) with p ∈ [1, ∞) in [3, 4], and f ∈ BV [0, 1] in [20]. By a refined version of the Riemann–Lebesgue lemma (cf. [16, Lemma 1.3.1]), the integral term in (1.1) is o(e |Im z| ) as |z|→∞, and Rouch´ e’s theorem shows that the functions F f and sin z have the same number of zeros in the discs |z|≤ π(l + 1 2 ) for all sufficiently large l ∈ N and that the zeros of F f approach those of sin z for Received by the editors September 26, 2006 and, in revised form, June 15, 2007. 2000 Mathematics Subject Classification. Primary 30D15; Secondary 42A38. Key words and phrases. Entire functions, asymptotics of zeros, Fourier transform. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 2207 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use