On zeros of some entire functions

We study the distribution of zeros $z_k$ for entire functions of the form $\sin z + \int_{0}^1 f(t)\mathrm{e}^{iz(1-2t)}\,dt$ with $f$ belonging to a space $X \hookrightarrow L_1(0,1)$. For a large class $\mathscr{X}$ of spaces $X$ (including, e.g., the spaces $L_p(0,1)$ for all $p\in[1,\infty]$) we show that $z_k=\pi k + \zeta_k$, where $(\zeta_k)_{k\in\mathbb{Z}}$ is the sequence of Fourier coefficients for some function $g$ in $X$, and study properties of the induced mapping $g\mapsto f$.