Lower bounds in $H^2$-rational approximation to Blaschke products

Abstract

We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose zeros are bounded away from the circle. The latter case depends on Fourier coefficients estimates for Blaschke products which are of independent interest.

Figures (4)

• Figure 1: Numerically optimal exponent $\beta_{\ast}(n;x)$ minimizing $K_{\beta}(n;x)=(n+1)^{\beta/2}\sqrt{S_{\beta}(x)}$ (with $S_{\beta}(x)=\sum_{m\ge0}x^m/(m+1)^{\beta}$ and $x=(s^*)^{-2}$) for representative values of $(\lambda,\alpha/\alpha_0)$. The optimizer decreases with $n$ and eventually reaches $0$, in agreement with the previous discussion.
• Figure 2: Comparison of the theoretical bound given in Theorem \ref{['initp']} and the best rational approximation found by RARL2 when approximating $z^{100}$ by a fraction of ${\cal R}_{n,n}$. The quantity plotted are actually one minus the bound and one minus the norm, so higher is better.
• Figure 3: Comparison of the theoretical bound given in Theorem \ref{['genBp']} and the best rational approximation found by RARL2 when approximating $B_1$, $B_2$ and $B_3$ by a fraction of ${\cal R}_{n,n}$. The quantity plotted are actually one minus the bound and one minus the norm, so higher is better.
• Figure 4: Values of the bound given by Theorem \ref{['genBp']} as a function of $\alpha$. For $\alpha \in (0,\alpha_0)$ on the left; zoom for $\alpha$ rather close to the point where the bound is maximal, on the right.

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Remark 1
• Theorem 3
• proof
• Remark 2