In this paper, we discuss “cardinal” as well as “infinite scattered data ” interpolation using finite linear combination of translates of a conditionally positive definite function. We prove that for a subclass of such functions, the cardinal interpolation operators are bounded and invertible on l _p, 1 ≤ ρ ≤ ∞. In some special cases, including $ldquo;Hardy's multiquadrics” and “The Thin Plate Spline,” we show that the scattered data interpolation operators are bounded and invertible on l ₂.