Polynomial approach to cyclicity for weighted $$\ell ^p_A$$

Abstract

In previous works, an approach to the study of cyclic functions in reproducing kernel Hilbert spaces has been presented, based on the study of so called optimal polynomial approximants. In the present article, we extend such approach to the (non-Hilbert) case of spaces of analytic functions whose Taylor coefficients are in ℓ p (ω), for some weight ω. When ω = {(k + 1) α } k∈N , for a fixed α ∈ R, we derive a characterization of the cyclicity of polynomial functions and, when 1 < p < ∞, we obtain sharp rates of convergence of the optimal norms.

Boundary Behavior of Optimal Polynomial Approximants

“…In order for the proofs here to go through, one needs good estimates, for |z| ≥ 1, on sums of the form N k=0 1 ω k z k that do not depend only on the modulus of z. (D) The approach discussed in this paper yields, in the forthcoming paper [17], the possibility of proving results on cyclicity and the corresponding rates of approximation on large classes of (non-Hilbert) Banach spaces. (E) Theorem 1.1 has not been established for functions g other than very particular polynomials of low degree (deg(g) ≤ deg(f )).…”

Zeros of optimal polynomial approximants in $\ell^p_{A}$

“…A and ℓ ∞ A are not uniformly convex, and hence the closest point to V is generally not attained; furthermore, when attained, the nearest point need not be unique. In fact, this failure of uniqueness is relevant to the study of optimal polynomial approximants [16].…”

“…When dealing with approximation theory in Banach spaces without a natural concept of orthogonality, a useful substitute is Birkhoff-James orthogonality. This generalization of Hilbert space orthogonality is valid in arbitrary normed spaces and has proven to be useful in exploring invariant subspaces and cyclicity in various Banach spaces of analytic functions [10,11,16].…”

“…The work in [4] deals with more general Hilbert spaces. Most of what we presented here can be extended with minimal changes to the context of weighted ℓ p A spaces (like the ones in [16], that is, defined by a norm f p p,ω := |a k | p ω k for some weight ω) but the nonlinearity of the recurrence relations is a key obstruction then and the dynamical systems approach of the previous section is not directly feasible.…”