We show that interpolation to a function, analytic on a compact set$E$in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution on$E$in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure on$E$, which may be different from the equilibrium measure.

Let$L$=−Δ+$V$be a Schrödinger operator on ℝ^{$d$},$d$≥3. We assume that$V$is a nonnegative, compactly supported potential that belongs to$L$^{$p$}(ℝ^{$d$}), for some$p$>$d$$/$2. Let$K$_{$t$}be the semigroup generated by −$L$. We say that an$L$^{1}(ℝ^{$d$})-function$f$belongs to the Hardy space $H^{1}_{L}$ associated with$L$if sup_{$t$>0}|$K$_{$t$}$f$| belongs to$L$^{1}(ℝ^{$d$}). We prove that $f\in H^{1}_{L}$ if and only if$R$_{$j$}$f$∈$L$^{1}(ℝ^{$d$}) for$j$=1,…,$d$, where$R$_{$j$}=($∂$/$∂$$x$_{$j$})$L$^{−1$/$2}are the Riesz transforms associated with$L$.

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