Exponential synthesis in the kernel of a $q$-sided convolution operator

The traditional approach to the problem of exponential synthesis for the space of solutions of a homogeneous convolution-type equation in a convex domain assumes that this space is invariant under some differential operator. This assumption makes it possible to reduce the problem of exponential synthesis to the problem of spectral synthesis. Is this assumption due to the method used to solve the problem, or the invariance of the solution space is necessary for a positive answer to the problem of exponential synthesis? To resolve this question the article considers special equations of the convolution type – the equations with $q$-sided convolution. It is shown that for such equations the requirement that the space of solutions be invariant is necessary and cannot be omitted if we assume that the solution space admits exponential synthesis with a free choice of the convex region and the characteristic function of the equation.