Splitting entire functions with zeros in a strip

The following result is proved. If $\varphi$ is a smooth function with support in the interval $[-N, N]$ and if all the zeros of its Fourier transform
$$\widehat\varphi(\lambda)=\int e^{\mathrm i\lambda t}\varphi(t)\,dt$$
are in some horizontal strip, then $\varphi$ can be represented as a convolution of two smooth functions with supports in the interval $[-N/2, N/2]$.