Fischer decomposition of the space of entire functions for the convolution operator

It is known that any function in a Hilbert Bargmann–Fock space can be represented as the sum of a solution of a given homogeneous differential equation with constant coefficients and a function being a multiple of the characteristic function of this equation with conjugate coefficients. In the paper, a decomposition of the space of entire functions of one complex variable with the topology of uniform convergence on compact sets for the convolution operator is presented. As a corollary, a solution of the de la Vallée Poussin interpolation problem for the convolution operator with interpolation points at the zeros of the characteristic function with conjugate coefficient is obtained.