Constructing polynomials of minimal growth

Abstract

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ _n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p _n ) such that lim _n→∞ p _n (λ _k ) = δ _j,k and lim sup _n→∞ sup _k>j |p _n (λ _k )|^1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.