Cyclicity in Dirichlet-type spaces and extremal polynomials

Abstract

For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\Vert pf-1{\Vert }_{\alpha }$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\Vert p_{n}f-1{\Vert }_{\alpha }$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.

26 pages, 2 figures, submitted for publication

References in corpus (2)

• Cyclicity in Dirichlet-type spaces and extremal polynomials
• Zeros of Sections of Some Power Series

Cited by in corpus (11)

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• Remarks on Inner Functions and Optimal Approximants
• Cyclicity in Dirichlet-type spaces and extremal polynomials II: functions on the bidisk
• Optimal approximants and orthogonal polynomials in several variables
• General Optimal Polynomial Approximants, Stabilization, and Projections of Unity
• Shift-cyclicity in analytic function spaces
• A note on Dirichlet-type spaces and cyclic vectors in the unit ball of ${\mathbb{C}}^2$
• More properties of optimal polynomial approximants in Hardy spaces
• Cyclicity in reproducing kernel Hilbert spaces of analytic functions
• Invariant subspaces of analytic perturbations