Mean growth of \(H^p\) functions (Q1281284)

Let \({\mathbf D}\) be the unit disk in the complex plane and \(A^{-\infty} ({\mathbf D})\) be the space of functions of polynomial growth. Sampling sets for \(A^{-\infty} ({\mathbf D})\) are those such that the restriction of a function in \(A^{-\infty} ({\mathbf D})\) to the set determines the type of growth of the function. The authors establish that sampling sets are always weakly sufficient and that weakly sufficient sets are always of uniqueness. It is shown that the converse implications do not hold.