Polynomial projectors preserving homogeneous partial differential equations

A polynomial projector Π of degree d on H(Cn) is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every f∈H(Cn) and every homogeneous polynomial of degree k, q(z)=∑|α|=kaαzα, there holds the implication: q(D)f=0⇒q(D)Π(f)=0. We prove that a polynomial projector Π preserves HPDE of degree k,1⩽k⩽d, if and only if there are analytic functionals μk,μk+1,…,μd∈H′(Cn) with μi(1)≠0,i=k,…,d, such that Π is represented in the following form Π(f)=∑|α|