Polynomial projectors preserving homogeneous partial differential equations

Abstract

A polynomial projector of degree d on H (C n ) is said to preserve homogeneous partial differential equations (HPDE) of degree k if for every f ∈ H (C n ) and every homogeneous polynomial of degree k, q(z) = | |=k a z , there holds the implication: q(D)f = 0 ⇒ q(D) (f ) = 0. We prove that a polynomial projector preserves HPDE of degree k, 1 k d, if and only if there are analytic functionals k , k+1 , . . . , d ∈ H (C n ) with i (1) = 0, i = k, . . . , d, such that is represented in the following formwith some a 's ∈ H (C n ), | | < k, where u (z) := z / !. Moreover, we give an example of polynomial projectors preserving HPDE of degree k (k 1) without preserving HPDE of smaller degree. We also give a characterization of Abel-Gontcharoff projectors as the only Birkhoff polynomial projectors that preserve all HPDE.