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

@inproceedings{Cheng2021ZerosOO, title={Zeros of optimal polynomial approximants in \$\ell^p\_\{A\}\$}, author={Raymond Cheng and William T. Ross and Daniel Seco}, year={2021} }

The study of inner and cyclic functions in lpA spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of lpA if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of p. We find the value of this radius for p 6= 2. In addition, for each positive integer d there is a…

## References

SHOWING 1-10 OF 20 REFERENCES

### Zeros of optimal polynomial approximants: Jacobi matrices and Jentzsch-type theorems

- MathematicsRevista Matemática Iberoamericana
- 2019

We study the structure of the zeros of optimal polynomial approximants to reciprocals of functions in Hilbert spaces of analytic functions in the unit disk. In many instances, we find the minimum…

### Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

- MathematicsJ. Lond. Math. Soc.
- 2016

It is shown that extremal polynomials are non-vanishing in the closed unit disk for $\alpha\in [0,1]$ (which includes the Hardy and Dirichlet spaces of the disk) and general $f$, and how this can be expressed in terms of quantities associated with orthogonal polynomers and kernels is explained.

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

- Mathematics
- 2020

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 \emph{optimal polynomial approximants}. In the…

### On the failure of canonical factorization in ℓAp

- MathematicsJournal of Mathematical Analysis and Applications
- 2019

### Beurling's Theorem for the Bergman space

- Mathematics
- 1996

A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in $H^2$ in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do…

### Shift Invariant Subspaces with Arbitrary Indices in ℓp Spaces

- Mathematics
- 2002

We construct right shift invariant subspaces of index n, 1⩽n⩽∞, in lp spaces, 2<p<∞, and in weighted lp spaces.

### Inner Functions in Reproducing Kernel Spaces

- MathematicsTrends in Mathematics
- 2019

In Beurling’s approach to inner functions for the shift operator S on the Hardy space H2, a function f is inner when f ⊥ Snf for all \(n \geqslant 1\). Inspired by this approach, this paper develops…

### On bounded analytic functions

- Mathematics
- 1950

The objective of this paper is to give an alternative derivation of results on bounded analytic functions recently obtained by Ahlfors [1] and Garabedian [2].1 While it is admitted that the main idea…