DSpace at VNU: Polynomial projectors preserving homogeneous partial differential equations

Journal of Approximation Theory 135 (2005) 221 – 232 www.elsevier.com/locate/jat Polynomial projectors preserving homogeneous partial differential equations Dinh-D˜unga,∗ , Jean-Paul Calvib , Nguyên Tiên Trunga a Information Technology Institute, Vietnam National University, Hanoi, E3, 144 Xuan Thuy Rd., Cau Giay, Hanoi, Vietnam b Laboratoire de Mathématiques E Picard, UFR MIG, Université Paul Sabatier, 31062 Toulouse Cedex, France Received 20 August 2004; accepted 21 April 2005 Communicated by Borislav Bojanav Available online 22 June 2005 Abstract A polynomial projector 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) = | |=k a z , there holds the implication: q(D)f = ⇒ q(D) (f ) = We prove that a polynomial projector preserves HPDE of degree k, k d, if and only if there are analytic functionals k , k+1 , , d ∈ H (Cn ) with i (1) = 0, i = k, , d, such that is represented in the following form (f ) = a (f )u + | |