Entire functions having a concordant value sequence

A classic theorem of Polya shows that 2(z) is, in a strong sense, the "smallest" transcendental entire function that is integer valued on N. An analogous result of Gel'fond concerns entire functions that are integer valued on the set X-a = {a(n): n is an element of N}, where a is an element of Z,\a\ greater than or equal to 2. Let X = N or X = X and k is an element of N or k = ∞. This paper pursues analogous results for entire functions f having the following property: on any finite subset D of X with #D less than or equal to k + 1, the values f (z), z is an element of D admit interpolation by an element of Z[z]. The results obtained assert that if the growth of f is suitably restricted then the restriction of f to X must be a polynomial. When X = X-a and k <∞ a "smallest" transcendental entire function having the requisite property is constructed.