On a space of entire functions decreasing rapidly on a real line

A space of entire functions decreasing rapidly on a real line is introduced. It contains the space of the Fourier–Laplace transforms of infinitely differentiable functions on a real line with a compact support as a proper subspace. The Fourier–Laplace transform of functions of this space is studied. Equivalent description in terms of estimates of derivatives of functions on a real line is obtained for the considered space.