Linear Interpolation on a Euclidean Ball in Double-struck capital R-n

摘要

For x((0)is an element of) R-n, R >0, by B = B(x((0)); R) we denote the Euclidean ball in R-n given by the inequality, parallel to x - x((0))parallel to <= R, parallel to x parallel to:= (Sigma(n)(i=1)x(i)(2))(1/2). Put B-n := B(0, 1). We mean by C(B) the space of continuous functions f : B -> R with the norm parallel to f parallel to(C(B)) := max (x is an element of B vertical bar)f(x)vertical bar and by Pi(1)(R-n) the set of polynomials in variables of degree <= 1, i. e., linear functions on R-n. Let x((1)), .... x((n+1)) be the vertices of dimensional nondegenerate simplex S subset of B. The interpolation projector P " C(B) -> Pi(1)(R)(n) corresponding to S is defined by the equalities Pf(x((i))) = f(x((j))) from C(B) into C(B). Let us define theta(n)(B) as minimal value of parallel to P parallel to(B) under the condition x((j)) is an element of B. In the paper we obtain the formula to compute parallel to P parallel to(B) making use of x((0)), R and coefficients of basic Lagrange polynomials of S. In more details we study the case when S is a regular simplex inscribed into B-n. In this situation, we prove that parallel to P parallel to(Bn) = max {psi(a), psi(a + 1)}where psi(t) = 2 root n/n +1(t(n + 1 - t))(1/2) + vertical bar 1 - 2t/n + 1 vertical bar (0 <= t <= n + 1) and integer has the form a = left perpendicular n + 1/2 - root n + 1/2 right perependicular. For this projector, root n <= parallel to P parallel to(Bn) <= root n + 1. The equality parallel to P parallel to(Bn) = root n + 1 takes place if root n + 1 and only if is an integer number. We give the precise values of theta(n)(B-n)for 1 <= n <= 4. To supplement theoretical results we present computational data. We also discuss some other questions concerning interpolation on a Euclidean ball.