Degenerate Orthogonality in the Polynomial Space Induced by Discrete and Hermite-Type Local Functionals

Abstract

This paper studies degenerate bilinear forms on the space of polynomials generated by discrete functionals of Lagrange and Hermite type. It is shown that for a discrete form built from polynomial values at a finite set of nodes, its kernel coincides with the principal ideal generated by the polynomial vanishing at all nodes. Next, a Hermite-type generalization is established: if the form depends on derivative values up to prescribed orders at the nodes, and the corresponding local coefficient matrices are nondegenerate, then the kernel of the form coincides with the principal ideal generated by the product of the corresponding powers of linear factors. This makes it possible to interpret the quotient space of polynomials modulo the kernel as a space of finite jets, and the corresponding orthogonal decomposition as Lagrange or Hermite interpolation. In addition, a general theorem on local forms on jet spaces is formulated, covering Lagrange, Hermite, and mixed local schemes within a unified framework.