Weighted holomorphic polynomial approximation

Abstract

For $G$ an open set in $\mathbb{C}$ and $W$ a non-vanishing holomorphic function in $G$, in the late 1990's, Pritsker and Varga characterized pairs $(G,W)$ having the property that any $f$ holomorphic in $G$ can be locally uniformly approximated in $G$ by weighted holomorphic polynomials $\{W(z)^np_n(z)\}, \ deg(p_n)\leq n$. We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs $(G,W)$. Then we consider the special case where $W(z)=1/(1+z)$ and $G$ is a loop of the lemniscate $\{z\in \mathbb{C}: |z(z+1)|=1/4\}$. We show the normalized measures associated to the zeros of the $n-th$ order Taylor polynomial about $0$ of the function $(1+z)^{-n}$ converge to the weighted equilibrium measure of $\overline G$ with weight $|W|$ as $n\to \infty$. This mimics the motivational case of Pritsker and Varga where $G$ is the inside of the Szego curve and $W(z)=e^{-z}$. Lastly, we initiate a study of weighted holomorphic polynomial approximation in $\mathbb{C}^n, \ n>1$.

Figures (4)

• Figure 1: A few level lines $E_{R}$, defined in (\ref{['def-ER']}), for the case of the segment $[1/2,1]$ and the weight $W(z)=z$.
• Figure 2: The lemniscate $4|z(z+1)|=1$.
• Figure 3: The 50 zeros of $s_{50}((1+z)^{-50})$, and the right loop $\mathcal{L}_{+}$ of the lemniscate $\mathcal{L}$, an analogue of the Szegö curve for the function $1/(1+z)$.
• Figure 4: Three level lines of $g(t)$ when $z=-0.4$, and the points $-1,z,0$. The level line $\mathcal{C}_{1}\cup\mathcal{C}_{2}$ with the inner loop $\mathcal{C}_{1}$ is the one passing through the critical point $t_{0}=2z+1=0.2$.

Theorems & Definitions (24)

• Definition 1.1
• Theorem 1.2: PV1
• Theorem
• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• Theorem 2.4
• proof
• Remark 2.5