Interpolation conditions and polynomial projectors preserving homogeneous partial differential equations
Let \(H(\mathbb C^n)\) be the space of entire functions on \(\mathbb C^n\). Let \({\mathcal P}_d(\mathbb C^n)\) be the space of all polynomials on \(\mathbb C^n\) of total degree \(\leq d\). A polynomial projector \(\Pi: H(\mathbb C^n) \to {\mathcal P}_d(\mathbb C^n)\) is said to preserve homogeneous partial differential equations of degree \(k\), if every homogeneous partial differential operator \[ q(D) = \sum_{| \alpha| =k} a_{\alpha}\,D^{\alpha} \] and for every \(f\in H(\mathbb C^n)\) with \(q(D)\,f =0\) it follows that \(q(D)\,\Pi(f) =0\). Examples of such polynomial projectors are Taylor, Abel-Goncharov, Kergin, and Hakopian projectors. In this paper, the author discusses some interpolation properties of polynomial projectors that preserve homogeneous partial differential equations.
projector preserving homogeneous partial differential equations
scientific article; zbMATH DE number 5152631