For a locally compact Hausdorff space \(X\) and a normed space \(E\) let \(C(X;E)\) denote the space of all continuous functions from \(X\) to \(E.\) A weight on \(X\) is a non-negative upper semi-continuous function \(v:X\to [0,\infty).\) For a directed family \(V\) of weights on \(X,\) one denotes by \(CV_\infty(X;E)\) the vector subspace of \(C(X;E)\) formed by all functions \(f\in C(X;E)\) such that \(vf\) vanishes at infinity for every \(v\in V.\) The family of seminorms \(\,p_v(f) =\sup\{v(x)\|f(x)\| : x\in X\},\; v\in V,\,\) generates a locally convex topology on \(CV_\infty(X;E)\). Let \(W\) be a nonempty subset of \(C(X;E).\) A multiplier for \(W\) is a function \(\phi\in C(X;[0,1])\) such that \(\phi f+(1-\phi) g\in W\) for every \(f,g\in W.\) The set of all multipliers is denoted by \(M(W).\) The family \(W\) is called interpolating for \(CV_\infty(X;E)\) if for every \(f\in CV_\infty(X;E)\) and every nonempty finite \(S\subset X\) there exists \(w\in W\) such that \(w(x)=f(x),\, x\in S.\) If further, for very \(f\in CV_\infty(X;E),\, \) every \(\epsilon > 0\) and every \(v\in V\) there exists \(w\in W\) interpolating \(f\) on \(S\) and such that \(p_v(f-w) < \epsilon,\) then one says that \(W\) has the property SAI (see F. Deutsch, [SIAM J. Appl. Math. 14, 1180-1190 (1966; Zbl 0173.06301)]).