Estimation of Interpolation Projectors Using Legendre Polynomials

We give some estimates for the minimal projector norm under linear interpolation on a compact subset of Rⁿ . Let Π ₁(Rⁿ) be the space of polynomials in n variables of degree at most $$1$$ , Ω is a compactum in Rⁿ , and K = conv(Ω ) . We will assume that vol(K) > 0 . Let x^(j) ∈ Ω , 1 j n + 1, be the vertices of an n -dimensional nondegenerate simplex. The interpolation projector P:C(Ω ) → Π ₁(Rⁿ) with the nodes x^(j) is defined by the equalities Pf( x^(j) ) = f( x^(j) ) . By \| P \|Ω we mean the norm of P as an operator from C(Ω ) to C(Ω ) . By θ ₙ(Ω ) we denote the minimal norm \| P \|Ω of all operators P with nodes belonging to Ω . Let simpₙ(Ω ) be the maximum volume of a simplex with vertices in Ω . We establish the inequalities χ ₙ^ - 1( vol(K)simpₙ(Ω ) ) θ ₙ(Ω ) n + 1. Here χ ₙ is the standardized Legendre polynomial of degree n . The lower estimate is proved using the obtained characterization of the Legendre polynomials through the volumes of convex polyhedra. More specifically, we show that for every γ 1 the volume of the set \ x = (x₁,...,xₙ) ∈ Rⁿ:∑ | xⱼ | + | 1 - ∑ xⱼ | γ \ is equal to χ ₙ(γ )/n! . In the case when Ω is an n -dimensional cube or an n -dimensional ball, the lower estimate gives the possibility to obtain the inequalities of the form θ ₙ(Ω ) c√ n . Also we formulate some open questions.