Weighted Polynomial Approximation on the Integers

We prove here some polynomial approximation theorems, somewhat related to the Szasz-Mfintz theorem, but where the domain of approximation is the integers, by dualizing a gap theorem of C. l ~ Y I for periodic entire functions. In another Paper [7], we shall prove, by similar means, a completeness theorem ibr some special sets of entire functions. I t is well known (see, for example [l]) tha t i f E is the space of all entire functions in the topology of uniform convergence on compact sets, then the dual space of continuous complex-valued linear functionals on E may be represented as E0, the space of entire functions of exponential type. Now let E (1) b e the space of entire functions of period 1. Then it may be shown that the dual of E (1) can be represented as E0 (l), where E0(1) is the following quotient space of E0: define / ~ g for functions 1, g e E0 i f / _ g is a multiple of sin z~z, and let E0 (l) be the space of equivalence classes of E0 modulo this relation of equivalence. Now E0(1) is apparent ly the same space as the space of restrictions of functions in E0 to the integers Z. Each such restriction is just a two-sided sequence of complex numbers, of at most exponential growth. ConVersely, it is easy to interpolate any such sequence by an entire function of exponential type. Thus, the dual of E(1) is just the space of all such sequences. Actually, we establish this identification by another procedure. To any theorem about periodic entire functions will correspond a theorem about the space of sequences described above. In [6], C. RI~NYI proved an interesting gap theorem, reproduced below. We show by means of duality tha t certain theorems of polynomial approximation are equivalent to this theorem. The domain of approximation is the integers in one case, and the positive integers in another ease. To our knowledge, the problem of polynomial approximation on the integers has not been Considered except in the note [3]. We know of no direct proof of our results. Ultimately, the R ~ Y I result depends on a simple application of Rolle's theorem. I t Would be of interest to have more precise gap theorems than the R~NYr theorem and also to have direct proofs of the results we prove by means of it.