A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms

TL;DR

This work extends weighted pluripotential theory to vector-valued polynomial interpolation by introducing a finite-dimensional Hermitian target space $U$, vector-valued weights, and a corresponding vector transfinite diameter. It establishes vector-valued Bernstein-Markov properties, Gram determinant asymptotics, and a vector analogue of strong Bergman asymptotics, showing convergence of Fekete vector measures to weighted equilibrium measures and expressing the diameter as a geometric mean of per-component scalar diameters. The theory is specialized to polynomial differential forms via $U=Lambda^k_$, introducing currents and unisolvent sets for form interpolation and highlighting vector Fekete currents as stable nodes. The framework provides a principled approach for interpolating tensor fields and differential forms with polynomial coefficients, with potential impact on numerical methods such as finite element schemes and structure-preserving interpolation.

Abstract

We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the proposed modification a vector valued weight is considered, allowing to partially extend the main asymptotic results holding for interpolation of scalar valued functions to the case of vector valued ones. As motivating example and main application we specialize our results to interpolation of differential forms by differential forms with polynomial coefficients.

Theorems & Definitions (14)

• Theorem 2.1
• proof
• Remark 3.1: BM vector measures asssociated to Fekete points
• Lemma 3.1
• Proposition 3.1
• proof
• Remark 4.1
• Proposition 4.1
• proof : Sketch of the proof
• Proposition 4.2