If the logistic kernel is a radial function, say g(r), it will reproduce a Hilbert space in a space variables of any dimensionality if and only if the univariate function f(t) = g(\sqrt{t}) is a completely monotonous function on (0,\infty) and is bounded at zero. Complete monotonity means that (-1)^m f^{(m)} is nonegative on (0,\nfty) for any m. In other words f should be nonnegative, f' should be nonpositive, f'' should be nonnegative, and so on. Excluding the case when f(t)=t^n for natural number n.

If the dimensionality of space of independent variables is limited (say 2-dimensional or 3-dimensional), it is enough that the function and its first derivatives satisfy the monotonity property.

See detail in Wedland, Holger, "Scattered Data Approximation", Cambridge Monographs on Applied and Computational Math., 2005, Chapters 7 and 8.