Uniqueness Classes for Difference Functionals

If ﹛Ln﹜ is a sequence of linear functionals on a linear space C of functions to the complex numbers, then a subspace C1 ⊂ C is a uniqueness class for ﹛Ln﹜ if a function f in C1 is uniquely determined by the sequence ﹛Ln(f)﹜ of complex numbers; i.e., if f ∈ C1 and Ln(f) = 0, n = 0, 1, 2, … , implies f = 0. For example, the class of all functions f analytic at the origin is a uniqueness class for the sequence ﹛f(n)(0)﹜ of linear functionals. Gontcharoff (9) asked the following question: Suppose, instead of ﹛f(n)(0)﹜, we use ﹛f(n)(an)﹜.