On a method for interpolating functions on chaotic nets

Abstract

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x|^m form odd,K(x)=|x|^m In |x| form even (x∈ℝⁿ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝⁿ. We construct a function of the form

$$\sum\limits_{j = 1}^N {\lambda _j K(x - x_j ) + P(x), where \lambda _j \in \mathbb{R}} , P \in \mathcal{P}, \sum\limits_{j = 1}^N {\lambda _j Q(x_j ) = 0 \forall Q \in } \mathcal{P}$$

coinciding with a given functionf(x) at the pointsx ₁,...,x _N . Error estimates for the approximation of functionsf∈W ^k _p (Ω) and theirlth-order derivatives in the normsL _q (Ω_ε) are obtained for this interpolation method, where Ω is a bounded domain in ℝⁿ, ε>0, and Ω_ε={x∈Ω:dist(x, ∂∈)>ε}.