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 $\|p f-1\|_\alpha$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\|p_{n}f-1\|_{\alpha}$ 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.Comment: 26 pages, 2 figures, submitted for publicatio

Orthogonal polynomials, reproducing kernels, and zeros of optimal approximants

“…The function g ≡ 1 is cyclic in all D α , and if a function f is cyclic in D α , then it is cyclic in D β for all β α. If f is a cyclic function, then the optimal approximants to 1/f have the property p n f − 1 α −→ 0, n −→ ∞, and the (p n ) yield the optimal rate of decay of these norms in terms of the degree n; see [3,5] for more detailed discussion of cyclicity. When α > 1, the algebra setting, cyclicity of f is actually equivalent to saying that f is invertible, but there exist smooth functions f that are cyclic in D α for α 1 without having 1/f ∈ D α : functions of the form f = (1 − z) N , N ∈ N, furnish simple examples.…”

“…When α > 1, the algebra setting, cyclicity of f is actually equivalent to saying that f is invertible, but there exist smooth functions f that are cyclic in D α for α 1 without having 1/f ∈ D α : functions of the form f = (1 − z) N , N ∈ N, furnish simple examples. In the paper [3], computations with optimal approximants resulted in the determination of sharp rates of decay of the norms p n f − 1 α for certain classes of functions with no zeros in the disk, but at least one zero on the unit circle T. Thus, the polynomials p n are useful and we deem them worthy of further study. A number of interesting questions arise naturally.…”

“…In [3,11], it was explained that (see [11,Theorem 2.1] for the particular statement used here) the coefficients (c k ) n k=0 of the nth optimal approximant are obtained by solving the linear system M c = e 0 , (1.4) with matrix M given by (M k,l ) n k,l=0 = ( z k f, z l f α ) k,l and e 0 = ( 1, f , . .…”

“…For simple functions f , this system can be solved in closed form for all n, leading to explicit expressions for p n . In [3], the authors found the optimal approximants to 1/(1 − z) for each α, and plotted their respective zero sets; a plot is reproduced in the next section. These plots, as well as the zero sets of optimal approximants for other simple functions f displayed in [3], all had one thing in common: the zeros of the polynomials p n , which we will denote by Z(p n ), were all outside the closed unit disk.…”

“…In [3], the authors found the optimal approximants to 1/(1 − z) for each α, and plotted their respective zero sets; a plot is reproduced in the next section. These plots, as well as the zero sets of optimal approximants for other simple functions f displayed in [3], all had one thing in common: the zeros of the polynomials p n , which we will denote by Z(p n ), were all outside the closed unit disk. Might this be true for any choice of f , or at least for f non-vanishing in the disk-are optimal approximants always zero-free in the disk?…”