Abstract
Multivariate polynomial interpolation plays a crucial role both in scientific computation and engineering application. Exploring the structure of the D-invariant (closed under differentiation) polynomial subspaces has significant meaning for multivariate Hermite-type interpolation (especially ideal interpolation). We analyze the structure of a D-invariant polynomial subspace Pn in terms of Cartesian tensors, where Pn is a subspace with a maximal total degree equal to n,n≥1. For an arbitrary homogeneous polynomial p(k) of total degree k in Pn, p(k) can be rewritten as the inner products of a kth order symmetric Cartesian tensor and k column vectors of indeterminates. We show that p(k) can be determined by all polynomials of a total degree one in Pn. Namely, if we treat all linear polynomials on the basis of Pn as a column vector, then this vector can be written as a product of a coefficient matrix A(1) and a column vector of indeterminates; our main result shows that the kth order symmetric Cartesian tensor corresponds to p(k) is a product of some so-called relational matrices and A(1).
Highlights
Multivariate polynomial interpolation is widely used in many application domains, such as image processing, electronic communication, control theory, etc
Note that most related practical problems can be converted to ideal interpolation problems, whose interpolation conditions is determined by a D −invariant polynomial subspace
In [5], by analyzing the structure of the second-order D-invariant subspaces, we give a sufficient condition to solve the discrete approximation problem for this case. This indicates that analyzing the structure of the D-invariant subspaces will help us know more about multivariate polynomial interpolation
Summary
Multivariate polynomial interpolation is widely used in many application domains, such as image processing, electronic communication, control theory, etc. In [5], by analyzing the structure of the second-order D-invariant subspaces, we give a sufficient condition to solve the discrete approximation problem for this case. This indicates that analyzing the structure of the D-invariant subspaces will help us know more about multivariate polynomial interpolation. We have seen that analyzing the structure of the D-invariant subspaces helps us to study a class of interpolation problems and to improve computational efficiency in polynomial system solving. It is the aim of this paper to describe the structure of the.
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