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The polynomial projectors that preserve homogeneous differential relations: a new characterization of Kergin interpolation. (English) Zbl 1113.41002

In the paper authors consider the polynomial projectors \({\mathbf \Pi}\) of degree \(d\) from the space of entire functions on \(\mathbb C^n\) to space of polynomials \(\mathcal P_d(\mathbb C^n)\), that preserve all polynomials and homogeneous differential relations (i.e., \(q(D)f=0 \rightarrow q(D)({\mathbf \Pi}(f))=0\), where \(q(D)=\sum_{|\alpha|=k}a_{\alpha}D^{\alpha}\) and \(D^{\alpha}= {\partial^k \over \partial z_1^{\alpha_1} \ldots \partial z_n^{\alpha_n}} \)). The main result shows that the projectors that preserve this homogeneous differential relations are the same ones that preserve ridge functions. For these projectors a characterization in term of the space of interpolation conditions is given. The authors also discuss the question of representing sequences, derivatives and the image of the projectors. In addition, the paper contains several applications to the study of the structure of the considered projectors, more precisely, to projectors that are related to Kergin interpolation.
Reviewer: Veselin Gushev

MSC:

41A05 Interpolation in approximation theory
41A63 Multidimensional problems
46A32 Spaces of linear operators; topological tensor products; approximation properties
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