Summary: For an interpolating Blaschke product \(b\),we denote by \(N(\bar b)\) and \(N_ 0(\bar b)\) the closures of the union sets of support sets of representing measures for points \(x\) in \(M(H^ \infty+ C)\) with \(| b(x)|< 1\) and \(b(x)=0\) respectively. Put \[ H^ \infty_{N(\bar b)}=\bigl\{ f\in L^ \infty;\;f_{| N(\bar b)}\in H^ \infty_{| N(\bar b)}\bigr\} \] and \[ A_ 0=\bigcap\bigl\{ H^ \infty_{\text{supp }\mu_ x};\;x\in M(H^ \infty+ C),\;b(x)= 0\bigr\}. \] It is proved that \(H^ \infty_{N(\bar b)}\subsetneq A_ 0\) if and only if there is a subproduct \(b_ 0\), such that \(N_ 0(\bar b_ 0)\subsetneq N(\bar b_ 0)\). We also discuss the relation between the conditions \(H^ \infty_{N(\bar b)}\subsetneq A_ 0\) and \(N_ 0(\bar b)\subsetneq N(\bar b)\).