Positive definiteness of a family of functions.

General necessary conditions on the real parameters α, β, C, D for the function where ρ is the norm on R, to be positive definite on R, are obtained. For ρ( x) = | x|, a criterion on these parameters is obtained in the following cases: (i) p = 1or p = 2; (ii) 3 < p ≤ ∞ and n = 2. [ABSTRACT FROM AUTHOR]