Problems and Results in the Calculation of Extremal Fundamental Systems for Sphere and Ball

Abstract

Let ℙ(D) denote any space of real polynomial restrictions onto the compact set D ⊂ ℝ^r, r ∈ ℕ \{1} with finite dimension N. The nodes t₁,...,t_N ∈ D are called a fundamental system (with regard to ℙ) if the corresponding evaluation-functionals are linear independent in ℙ′. In this case the Lagrangians L₁,...,L_N ∈ ℙ are well-defined by

$${{L}_{j}}({{t}_{k}})={{\delta }_{j,k}}for j,k\in \{1,\ldots N\}$$