Experimental Approaches to Kuttner's Problem

Abstract

For λ $\in$(0, 2) let k(λ) denote the smallest positive value of κ so that the truncated power function φ _{λ, κ} (t) = (1 -- |t|^λ)^κ₊ is positive definite. We give lower and upper estimates of Kuttner's function k(λ) through detailed numerical and symbolic computations, and we show analytically that k((4n+1)/(2n+1)) ≤ 2n+1 for n $\in$ N.