| Sbornik: Mathematics |
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Polynomial approximation on parabolic manifolds A. Sadullaeva, A. A. Atamuratovb a National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan b V. I. Romanovskiy Institute of Mathematics of the Academy of Sciences of Uzbekistan, Tashkent, Uzbekistan
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     § 1. IntroductionIn the classification of Riemann surfaces, ones on which there exist no nonconstant bounded above subharmonic function are called parabolic Riemann surfaces. In the general case, for complex manifolds of arbitrary dimension, various definitions of a parabolic manifold are known. A multidimensional parabolic manifold on which a special plurisubharmonic exhaustion function is assumed to exist was considered by Griffith and King [1] and Stoll [2], [3], who applied the properties of parabolic manifolds to multidimensional Nevanlinna theory. Griffith and King concentrated on affine algebraic subvarieties of the complex space. The Valiron defect divisors of holomorphic maps of parabolic manifolds were considered by the first-named author in [4]. The further development of the theory of holomorphic and plurisubharmonic functions on parabolic manifolds was connected with [5]–[16]. Here we use the following definitions, borrowed from joint papers of the first-named author with Aytuna (see [14], § 2). Definition 1.1. A Stein manifold $X$ is said to be parabolic if it carries no bounded above plurisubharmonic functions other than constants (that is, a plurisubharmonic function on $X$ which is bounded above is a constant). Definition 1.2. A Stein manifold $X$ of dimension $n$ is called an $S$-parabolic manifold if it carries a special exhaustion function $\rho (z)$: If, in addition, $\rho (z)\in C(X)$, then $X$ is called an $S^{*}$-parabolic manifold. Remark 1.1. For Riemann surfaces ($\dim X=1$) the notions of a parabolic, an $S$-parabolic and an $S^{*}$-parabolic manifold coincide (see [17], Ch. V, § 3.13A). For $\dim X>1$ this question is still open. We can introduce polynomials on $S$-parabolic manifolds Definition 1.3. Given a function $f(z)\in {\mathcal O}(X)$, if there exist positive numbers $c$ and $d$ such that for all $z\in X$ where $\rho^{+}(z)=\max \{0,\rho(z)\}$, then $f$ is called a $\rho$-polynomial of degree $\leqslant d$. The least possible value of $d$ in (1.1) is called the degree of the polynomial $f$. For fixed $d>0$ let ${\mathcal P}_{\rho}^{d} (X)$ denote the space of $\rho$-polynomials of degree $\leqslant d$ and ${\mathcal P}_{\rho}(X)=\bigcup_{d=0}^{\infty}{\mathcal P}_{\rho}^{d} (X)$ denote the set of all $\rho$-polynomials on $X$. It can be proved that for any $S$-parabolic manifold $X$ the vector space ${\mathcal P}_{\rho}^{d}(X)$ has a finite dimension; it has the upper estimate (see [14], Theorem 3.4, and [7], Theorem 4.8) where $n$ is the dimension of $X$.The simplest example of a parabolic manifold is the space $\mathbb{C}^{n}$ with exhaustion function $\rho=\log|z|$. Moreover, each algebraic manifold is parabolic. In these cases a rich class of polynomials $p(z)$ exists (Example 2.2). However, there is an example (see Example 2.1) of a parabolic manifolds without nontrivial polynomials other than constants. § 2. Two examplesFor completeness we present an example of a parabolic manifold on which there are no nontrivial polynomials other than constants. For details of the proof of the existence of such a manifold, see the joint paper [14] (Theorem 4.1) by the first-named author and Aytuna. The construction is based on the existence of a certain compact set $K\subset {\mathbb C}$ and a special potential $\displaystyle U^{\mu}(z)=\int \log|z-\xi|\,d\mu(\xi)$, $\operatorname{supp} \mu \subset K$, related to $K$. Theorem 2.1. There exist a polar compact set $K\subset {\mathbb C}$ and a subharmonic function $u(z)$ on ${\mathbb C}$, which is harmonic in ${\mathbb C}\setminus K$, satisfies $u|_{K}=-\infty$ and tends to $ -\infty $ slower than $ \log \operatorname{dist} (z,K)$, such that Proof. We list the main steps of the construction of the potential $U^{\mu}(z)$.
1. Choosing the compact set $K\subset {\mathbb C}$. We take it in the form of a special Cantor set $K\subset [0,1]\subset {\mathbb C}$. We choose the probability measure $\mu$, $\operatorname{supp}\mu \subset K$, so that the potential $U^{\mu}(z)$ of $\mu$ satisfies (2.1). We consider the interval $[0,1]$ and denote it by $K_{0}=[a_{01}, b_{01}]$; the length of $K_{0}$ is $1$. We construct the Cantor set as follows. Fix $\delta=1/4$ and the sequence $t_{m}=4^{m-1}$, $m=1,2,\dots $ . From the interval $[a_{01}, b_{01}]$ we remove the subinterval $(a_{01} +\delta, b_{01} -\delta)$ and obtain the union of two closed intervals $K_{1}=[a_{01}, a_{01} +\delta]\cup [a_{02} -\delta, a_{02}]$, which we re-denote by
$$
\begin{equation*}
K_{1}=[a_{01},a_{01} +\delta ]\cup [b_{01} -\delta,b_{01} ]=[a_{11},b_{11} ]\cup [a_{12},b_{12} ].
\end{equation*}
\notag
$$
These closed interval have lengths
$$
\begin{equation*}
|b_{11} -a_{11} |=\delta\quad\text{and} \quad | b_{12} -a_{12} |=\delta, \quad\text{and} \quad |b_{11} -a_{12} |=1-2\delta .
\end{equation*}
\notag
$$
Now we proceed the same way for each of these two intervals, but replace $\delta $ by $\delta^{t_{2}}$. Then we obtain a union of four closed intervals, of length $\delta^{t_{2}}$ each:
$$
\begin{equation*}
\begin{aligned} \, K_2 &=[a_{11}, a_{11} +\delta^{t_{2}} ]\cup [b_{11} -\delta^{t_{2}}, b_{11} ]\cup [a_{12}, a_{12} +\delta^{t_{2}} ]\cup [b_{12} -\delta^{t_{2}}, b_{12} ] \\ &=[a_{21}, b_{21}]\cup [a_{22}, b_{22} ]\cup [a_{23}, b_{23} ]\cup [a_{24}, b_{24} ], \end{aligned}
\end{equation*}
\notag
$$
and the distances between their endpoints are
$$
\begin{equation*}
|b_{21} -a_{22} |=\delta -2\delta^{t_{2}}, \qquad |b_{22} -a_{23} |=1-2\delta\quad\text{and} \quad |b_{23} -a_{24} |=\delta -2\delta^{t_{2}}.
\end{equation*}
\notag
$$
Repeating this procedure $m$ times we obtain a union of $2^{m}$ closed intervals of length $\delta^{t_{m}}$ each:
$$
\begin{equation*}
K_{m}=[a_{m1}, b_{m1} ]\cup [a_{m2}, b_{m2} ]\cup \dots \cup [a_{m2^{m}}, b_{m2^{m}} ].
\end{equation*}
\notag
$$
Note that $K_{0} \supset K_{1} \supset\dots \supset K_{m}\supset \dotsb$ and the Lebesgue measure $l(K_{m})$ is equal to $2^{m} \delta^{t_{m}}$. Furthermore, the Hausdorff measure with kernel $h(s)={1}/{\log(1/s)}$ of $K_{m}$ is
$$
\begin{equation}
H^{h} (K_{m})=2^{m} h\biggl(\frac{\delta^{t_{m}}}2\biggr) =2^{m} \biggl(\log \frac{1}{\delta^{t_{m}} /2}\biggr)^{-1} =\frac{2^{m}}{t_{m} \log(2^{1/t_{m}}/\delta)}.
\end{equation}
\tag{2.2}
$$
Set $K=\bigcap_{m=1}^{\infty}K_{m}$. It follows from the inequality $2^{m}/t_{m} \leqslant C<\infty$, $m=1,2,\dots$, that $H^{h} (K)<\infty$, and by a well-known property of logarithmic capacity the capacity $C(K)$ is zero. Hence there exists a probability measure $\mu$, $\operatorname{supp} \mu=K$, such that the potential is harmonic outside $K$, subharmonic in ${\mathbb C}$ and $U^{\mu} (z)=-\infty$ $\forall\,z\in K$.
Now we must construct a special measure $\mu$ satisfying (2.1). For $K_{m}=[a_{m1},b_{m1}]\cup [a_{m2}, b_{m2}]\cup\dots\cup[a_{22^{m}},b_{22^{m}}]$ set
$$
\begin{equation}
\mu_{m}=\frac{\delta (a_{m1})+\dots +\delta (a_{m2^{m}})+\delta (b_{m1})+\dots +\delta (b_{m2^{m}})}{2\cdot 2^{m}},
\end{equation}
\tag{2.3}
$$
where $\delta (c)$ is the unit measure concentrated at the point $c$. The sequence $\mu_{m}$ converges weakly to some measure: $\mu_{m}\to\mu$, ${\operatorname{supp}}\mu=K$. Let be the potential of $\mu$.
2. Estimates for the potential. We have $\lim_{z\to K}\mkern-1mu U^{\mu}(\mkern-1mu z\mkern-1mu)\mkern-1mu\!=\!-\infty$, that is, ${U^{\mu} |_{K}\!=\!-\infty}$. Consider a point $z^{0} \in{\mathbb C}\setminus K$, and let $\lambda=\operatorname{dist} (z^{0}, K)>0$. Then by the well-known integral formula (see [18], Ch. 2, § 2.5.18)
$$
\begin{equation}
U^{\mu_{m}} (z^{0})=\int \log|z^{0}-w|\,d\mu_{m}(w) =\int_{0}^{\infty}[\log t]\,d\mu_{m}(z^{0},t) =\int_{\lambda_{m}}^{\Lambda}[\log t]\,d\mu_{m}(z^{0},t),
\end{equation}
\tag{2.4}
$$
where $\mu_{m} (z^{0},t)=\mu_{m}(B(z^{0},t))$, $B(z^{0},t):=\{z\colon |z-z^{0}|\leqslant t\}$ is a closed disc, $\Lambda=\max\{\operatorname{dist} (z^{0}, 0), \operatorname{dist} (z^{0}, 1)\}$, $\lambda_{m}=\min\{|z^{0} -a_{mj}|,|z^{0} -b_{mj}|\colon j=1,2,\dots,2^{m} \}$ is the distance of $z^{0}$ to the set $K_{m}^{\mathrm{knot}}=\{a_{m1}, b_{m1}, a_{m2},b_{m2},\dots, a_{m2^{m}}, b_{m2^{m}} \}$ of boundary points, and $\lambda_{m} \geqslant \lambda$. Integrating by parts in (2.4) we obtain
$$
\begin{equation*}
\begin{aligned} \, U^{\mu_{m}}(z^0) &=\int_{\lambda_m}^{\Lambda}[\log t]\,d\mu_m(z^0,t) =\mu_m(z^0,t)\log t|_{\lambda_m}^{\Lambda}-\int_{\lambda_m}^{\Lambda}\frac{\mu_m(z^0,t)}{t}\,dt \\ &=\log \Lambda -\int_{\lambda_m}^{\Lambda}\frac{\mu_m(z^0,t)}{t}\,dt. \end{aligned}
\end{equation*}
\notag
$$
Now we estimate the potentials $U^{\mu_{m}}(z^{0})$ at points $z^{0}$ close to $K$, namely, for $\lambda_{m} <1$. Let $c$ be a boundary point of $K$ such that $\lambda_{m}=|z^{0} -c|$. The simplest cases are $c=0$ and $c=1$, but all other cases can be reduced to these two by dividing the set of boundary points ${a_{m1}, b_{m1}, a_{m2}, b_{m2},\dots, a_{m2^{m}},b_{m2^{m}}}$ into two groups in accordance with their position on the left or right of the point $\operatorname{Re} z^{0}$. Thus we can assume without loss of generality that $c=0$ and $\operatorname{Re} z^{0} \leqslant 0$. In this case
$$
\begin{equation*}
\mu_{m}(0, t-\lambda_{m})\leqslant \mu_{m} (z^{0}, t)\leqslant \mu_{m} (0,\sqrt{t^{2} -\lambda_{m}^{2}}).
\end{equation*}
\notag
$$
Setting $\mu_{m}(t)=\mu_{m} (0, t)$ we obtain
$$
\begin{equation}
-\int_{\lambda_{m}}^{\Lambda} \frac{\mu_{m} (t-\lambda_{m})}{t} \,dt \leqslant -\int_{\lambda_{m}}^{\Lambda} \frac{\mu_{m} (z^{0}, t)}{t} \,dt \leqslant-\int_{\lambda_{m}}^{\Lambda} \frac{\mu_{m}(\sqrt{t^{2} -\lambda_{m}^{2}})}{t} \,dt.
\end{equation}
\tag{2.5}
$$
It is obvious that
$$
\begin{equation*}
\mu_{m} (\delta)=\frac{1}{2}, \quad \mu_{m} (\delta^{t_{2}})=\frac{1}{2^{2}}, \quad \dots, \quad \mu_{m} (\delta^{t_{m-1}})=\frac{1}{2^{m-1}}\quad\text{and} \quad \mu_{m}(\delta^{t_{m}})=\frac{1}{2^{m}}.
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\begin{gathered} \, \mu_m(t) =\frac{1}{2}\quad\text{if}\ \delta\leqslant t <1-\delta, \\ \mu_m(t)=\frac{1}{2^2} \quad\text{if}\ \delta^{t_{2}}\leqslant t <\delta-\delta^{t_{2}}, \\ \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots\dots \\ \mu_m(t)=\frac{1}{2^{m-1}} \quad\text{if}\ \delta^{t_{m-1}}\leqslant t <\delta^{t_{m-2}}-\delta^{t_{m-1}}, \\ \mu_m(t)=\frac{1}{2^{m}} \quad\text{if}\ \delta^{t_{m}}\leqslant t <\delta^{t_{m-1}}-\delta^{t_{m}}. \end{gathered}
\end{equation}
\tag{2.6}
$$
Using inequality (2.5) and equality (2.6) we can find upper and lower bounds for $U^{\mu}(z)$. We have
$$
\begin{equation}
\begin{aligned} \, \notag I_m &=-\int_{\lambda_m}^{\Lambda}\frac{\mu_m(z^0,t)}{t}\,dt \leqslant -\int_{\lambda_m}^{\Lambda}\frac{\mu_m(\sqrt{t^2-\lambda_m^2})}{t}\,dt \\ &\notag =-\int_{0}^{\sqrt{\Lambda^2-\lambda_m^2}}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt -\int_{0}^{\delta^{t_{m-1}}-\delta^{t_{m}}}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt \\ \notag &\qquad -\int_{\delta^{t_{m-1}}-\delta^{t_{m}}}^{\delta^{t_{m-2}}-\delta^{t_{m-1}}}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt -\dotsb -\int_{\delta-\delta^{t_{2}}}^{1-\delta}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt \\ \notag &\qquad -\int_{1-\delta}^{1}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt -\int_{\delta^{t_{m}}}^{\delta^{t_{m-1}}-\delta^{t_{m}}} \frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt \\ \notag &\qquad -\int_{\delta^{t_{m-1}}}^{\delta^{t_{m-2}}-\delta^{t_{m-1}}} \frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt -\dots -\int_{\delta}^{1-\delta}\frac{t}{t^2+\lambda_m^2}\mu_m(t)\,dt \\ \notag &=-\frac{2}{2^{m+1}} \int_{\delta^{t_{m}}}^{\delta^{t_{m-1}} -\delta^{t_{m}}}\frac{t\,dt}{t^2+\lambda_m^2} -\frac{2^2}{2^{m+1}}\int_{\delta^{t_{m-1}}}^{\delta^{t_{m-2}}-\delta^{t_{m-1}}} \frac{t\,dt}{t^2+\lambda_m^2} -\dotsb \\ &\qquad -\frac{2^m}{2^{m+1}} \int_{\delta}^{1-\delta}\frac{t\,dt}{t^2+\lambda_m^2} . \end{aligned}
\end{equation}
\tag{2.7}
$$
Therefore,
$$
\begin{equation}
\begin{aligned} \, \notag I_m &\leqslant \frac{2}{2^{m+2}}\log \frac{\lambda_m^2+\delta^{2t_{m}}}{\lambda_m^2+(\delta^{t_{m-1}}-\delta^{t_{m}})^2} +\frac{2^2}{2^{m+2}}\log \frac{\lambda_m^2+\delta^{2t_{m-1}}}{\lambda_m^2+(\delta^{t_{m-2}}-\delta^{t_{m-1}})^2} +\dotsb \\ \notag &\qquad +\frac{2^m}{2^{m+1}}\log \frac{\lambda_m^2+\delta^2}{\lambda_m^2+(1- \delta)^2} \\ \notag &=\frac{1}{2^{m}}\log\frac{\lambda_m^2+\delta^{2t_{m}}}{\lambda_m^2+\delta^{2t_{m-1}}} +\frac{2}{2^{m}}\log \frac{\lambda_m^2+\delta^{2t_{m-1}}}{\lambda_m^2+\delta^{2t_{m-2}}} +\dotsb +\frac{2^{m-1}}{2^{m}}\log \frac{\lambda_m^2+\delta^2}{\lambda_m^2+1} \\ \notag &\qquad +o(\delta^{t_{m-1}}) \\ \notag &=\frac{1}{2^{m}}\log (\lambda_m^2+\delta^{2t_{m}})\frac{1}{2^{m}}\log (\lambda_m^2+\delta^{2t_{m-1}}) +\frac{2}{2^{m}}\log (\lambda_m^2+\delta^{2t_{m-2}})+\dotsb \\ &\qquad +\frac{2^{m-2}}{2^{m}} \log (\lambda_m^2+\delta^2) -\frac{2^{m-1}}{2^{m}}\log (\lambda_m^2+1)+o(\delta^{t_{m-1}}). \end{aligned}
\end{equation}
\tag{2.8}
$$
Let $k=k(z^{0})$ be the least positive integer such that $\delta^{t_{k}} \leqslant \lambda_{m}$. We split the last sum in (2.8) into two: over $k\leqslant j\leqslant m$ ($\delta^{t_{j}} \leqslant \lambda_{m}$) and over $j<k$ ($\delta^{t_{j}} >\lambda_{m}$). For the first sum, when $\delta^{t_{j}}\leqslant\lambda_{m}$, we can write
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{2^m}\log (\lambda_m^2+\delta^{2t_{m}}) +\frac{1}{2^m}\log (\lambda_m^2+\delta^{2t_{m-1}}) +\frac{2}{2^m}\log (\lambda_m^2+\delta^{2t_{m-2}})+\dotsb \\ &\qquad\qquad +\frac{2^{m-k-1}}{2^m}\log (\lambda_m^2+\delta^{2t_{m}}) \\ &\qquad \leqslant \frac{1+2+\dots +2^{m-k-1}}{2^m}\log (2\lambda_m^2) =\frac{2^{m-k}-1}{2^m}\log(2\lambda_m^2)\leqslant \frac{1}{2^m}\log (2\lambda_m^2). \end{aligned}
\end{equation*}
\notag
$$
Since $t_{k}=4^{k-1}$ and $\delta^{t_{k}} \leqslant \lambda_{m}$, it follows that $2^{k} \geqslant \sqrt{{\log\lambda_{m}}/{\log\delta}}$. Hence the first sum is at most
$$
\begin{equation*}
\frac{1}{2^{k}} \log(2\lambda_{m}^{2}) \leqslant \sqrt{\log\frac{1}{\delta}}\, \frac{2\log\lambda_{m} +\log2}{\sqrt{\log({1}/{\lambda_{m}})}}.
\end{equation*}
\notag
$$
For the second sum, when $\delta^{t_{j}} >\lambda_{m}$, we have
$$
\begin{equation*}
\begin{aligned} \, &\frac{2^{m-k}}{2^{m}} \log(\lambda_{m}^{2} +\delta^{2t_{k-1}})+\dots +\frac{2^{m-2}}{2^{m}} \log(\lambda_{m}^{2} +\delta^{2})-\frac{2^{m-1}}{2^{m}} \log(\lambda_{m}^{2} +1) \\ &\qquad \leqslant -\frac{1}{2} \log(\lambda_{m}^{2} +1)+\frac{1}{2^{2}} \log(\lambda_{m}^{2} +\delta^{2})+\dots +\frac{1}{2^{k}} \log(\lambda_{m}^{2}+\delta^{2t_{k-1}}) \\ &\qquad \leqslant-\frac{1}{2}\log(\lambda_{m}^{2}+1)+\frac{1}{2^{2}} \log(2\delta^{2})+\dots +\frac{1}{2^{k}}\log(2\delta^{2t_{k-1}}) \\ &\qquad \leqslant -\frac{1}{2} \log(\lambda_{m}^{2} +1)+\frac{1}{2^{2}}\log(2\delta^{2})+\dots +\frac{1}{2^{k+1}} \log(2\delta^{2}) \\ &\qquad =-\frac{1}{2} \log(\lambda_{m}^{2} +1)+\frac{1}{2}\log(2\delta^{2}). \end{aligned}
\end{equation*}
\notag
$$
Hence for sufficiently large $m$ we have the estimate
$$
\begin{equation}
U^{\mu_{m}}(z^{0})\leqslant \sqrt{\log\frac{1}{\delta}} \, \frac{\log\lambda_{m} +\log2}{\sqrt{\log({1}/{\lambda_{m}})}} -\frac{1}{2} \log(\lambda_{m}^{2} +1)+\log\Lambda +\frac{1}{2} \log(2\delta) +o(\delta^{t_{m-1}}).
\end{equation}
\tag{2.9}
$$
For at arbitrary $z^{0} \in {\mathbb C}\setminus K$ estimate (2.9) looks like
$$
\begin{equation}
\begin{aligned} \, \notag U^{\mu_{m}}(z^{0}) &\leqslant 2\sqrt{\log\frac{1}{\delta}}\, \frac{\log\operatorname{dist} (z^{0}, K_{m}^{\mathrm{knot}})+\log2}{\sqrt{\log(1/\operatorname{dist}(z^{0}, K_{m}^{\mathrm{knot}}))}} \\ &\quad-\frac{1}{2}\log\bigl(\operatorname{dist}^{2} (z^{0}, K_{m}^{\mathrm{knot}})+1\bigr)+\log\Lambda +\frac{1}{2} \log(2\delta) +o(\delta^{t_{m-1}}). \end{aligned}
\end{equation}
\tag{2.10}
$$
Letting $m$ tend to infinity in (2.10) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag & U^{\mu}(z^{0})\leqslant 2\sqrt{\log\frac{1}{\delta}}\,\frac{\log \operatorname{dist} (z^{0}, K)+\log2}{\sqrt{(1/\operatorname{dist}(z^{0}, K))}} \\ &\qquad\qquad -\frac{1}{2} \log (\operatorname{dist}^{2} (z^{0}, K)+1)+\log\Lambda +\frac{1}{2} \log(2\delta), \end{aligned}
\end{equation}
\tag{2.11}
$$
which shows, in particular, that $U^{\mu}(z^{0})=-\infty$ $\forall\, z^{0} \in K$.
3. The rate of convergence of $U^{\mu} (z)$ to $-\infty$ as ${z\to K}$. We have
$$
\begin{equation*}
\begin{aligned} \, I_m &=-\int_{\lambda_m}^{\Lambda}\frac{\mu_m(z^0,t)}{t}\,dt \geqslant -\int_{\lambda_m}^{\Lambda}\frac{\mu_m(t-\lambda_m)}{t}\,dt= -\int_{0}^{\Lambda-\lambda_m}\frac{\mu_m(t)}{t+\lambda_m}\,dt \\ &=-\int_{0}^{\delta^{t_{m-1}}-\delta^{t_{m}}}\frac{\mu_m(t)}{t+\lambda_m}\,dt -\int_{\delta^{t_{m-1}}-\delta^{t_{m}}}^{\delta^{t_{m-2}} -\delta^{t_{m-1}}}\frac{\mu_m(t)}{t+\lambda_m}\,dt -\dotsb \\ &\qquad -\int_{\delta-\delta^{t_{2}}}^{1-\delta}\frac{\mu_m(t)}{t+\lambda_m}\,dt -\int_{1-\delta}^{1}\frac{\mu_m(t)}{t+\lambda_m}\,dt \\ &\geqslant -\frac{2}{2^{m+1}} \int_{0}^{\delta^{t_{m-1}}-\delta^{t_{m}}} \frac{dt}{t+\lambda_m} -\frac{2^2}{2^{m+1}} \int_{\delta^{t_{m-1}}- \delta^{t_{m}}}^{\delta^{t_{m-2}}-\delta^{t_{m-1}}}\frac{dt}{t+\lambda_m} -\dotsb \\ &\qquad -\frac{2^m}{2^{m+1}} \int_{\delta-\delta^{t_2}}^{1-\delta}\frac{dt}{t+\lambda_m} -\frac{2^{m+1}}{2^{m+1}} \int_{1-\delta}^{1}\frac{dt}{t+\lambda_m} \\ &=-\frac{1}{2^{m}}\log \frac{\lambda_m+\delta^{t_{m-1}}-\delta^{t_{m}}}{\lambda_m} -\frac{1}{2^{m-1}}\log \frac{\lambda_m+\delta^{t_{m-2}}-\delta^{t_{m-1}}} {\lambda_m+\delta^{t_{m-1}}-\delta^{t_{m}}}-\dotsb \\ &\qquad -\frac{1}{2}\log \frac{\lambda_m+1-\delta}{\lambda_m+\delta-\delta^{t_{2}}} -\log \frac{ \lambda_m+1}{\lambda_m+1-\delta} \\ &=\frac{\log\lambda_m}{2^m} +\frac{\log (\lambda_m+\delta^{t_{m-1}}-\delta^{t_{m}})}{2^m} +\frac{\log(\lambda_m+ \delta^{t_{m-2}}-\delta^{t_{m-1}})}{2^{m-1}}+\dotsb \\ &\qquad +\frac{\log(\lambda_m+1-\delta)}{2}-\log(\lambda_m+1). \end{aligned}
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation*}
\begin{aligned} \, I_{m} &\geqslant -\log(\lambda_{m} +1)+\frac{\log(\lambda_{m} + 1-\delta)}{2} +\frac{\log(\lambda_{m} + \delta -\delta^{t_{2}})}{2^{2}} \\ &\qquad+\frac{\log(\lambda_{m} + \delta^{t_{2}} -\delta^{t_{3}})}{2^{3}} +\dotsb +\frac{\log(\lambda_{m} +\delta^{t_{m-2}} -\delta^{t_{m-1}})}{2^{m-1}} \\ &\qquad+\frac{\log(\lambda_{m} +\delta^{t_{m-1}} -\delta^{t_{m}})}{2^{m}}\cdot \frac{\log\lambda_{m}}{2^{m}} \\ &\geqslant -\log(\lambda_{m} +1)+\frac{\log(1-\delta)}{2} +\frac{\log(\delta -\delta^{t_{2}})}{2^{2}} +\dots +\frac{\log(\delta^{t_{k-1}} -\delta^{t_{k}})}{2^{k}} \\ &\qquad +\frac{\log\lambda_{m}}{2^{k+1}} +\dots +\frac{\log\lambda_{m}}{2^{m-1}} +\frac{\log\lambda_{m}}{2^{m}} +\frac{\log\lambda_{m}}{2^{m}} \\ &=c(k)+\frac{\log\lambda_{m}}{2^{k}}\cdot\biggl(1-\frac{1}{2^{m-k}}\biggr), \end{aligned}
\end{equation*}
\notag
$$
where $c(k)=\mathrm{const}$ is independent of $m$. Hence for each fixed $k\in {\mathbb N}$ we have
$$
\begin{equation}
U^{\mu_{m}}(z^{0})\geqslant \log\Lambda +c(k)+\frac{\log\lambda_{m}}{2^{k}}\cdot \biggl(1-\frac{1}{2^{m-k}}\biggr).
\end{equation}
\tag{2.12}
$$
Thus, for an arbitrary $z^{0} \notin K$,
$$
\begin{equation}
U^{\mu_{m}}(z^{0})\geqslant 2\biggl(\log\Lambda +c(k)+\frac{\log\operatorname{dist} (z^{0}, K_{m}^{\mathrm{knot}})}{2^{k}}\cdot \biggl(1-\frac{1}{2^{m-k}}\biggr)\biggr).
\end{equation}
\tag{2.13}
$$
Letting $m$ tend to infinity, we conclude from (2.13) that for each positive $\varepsilon$ there exists a constant $c(\varepsilon)>-\infty$ such that
$$
\begin{equation*}
U^{\mu}(z^{0})\geqslant c(\varepsilon)+\varepsilon\cdot \log\operatorname{dist} (z^{0}, K) \quad \forall\, z^{0} \in {\mathbb C}.
\end{equation*}
\notag
$$
Hence we obtain
$$
\begin{equation*}
\frac{U^{\mu}(z^{0})}{\log \operatorname{dist} (z,K)} \leqslant \varepsilon,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\lim_{z\to K} \frac{U^{\mu} (z^{0})}{\log \operatorname{dist} (z,K)}=0.
\end{equation*}
\notag
$$
Theorem 2.1 is proved. Now we can construct our example. Example 2.1. Consider the manifold $X=\overline{\mathbb C}\setminus K$, where $K$ is the compact set constructed above. As the special exhaustion function we take the potential $\rho(z)=-U^{\mu}(z)$. Then $\rho(z)$ is harmonic in $X\setminus\{\infty\}$, $\rho(\infty)=-\infty$ and $\rho (z)\to \infty$ as $z\to K$. Thus, $(X,\rho)$ is an $S^{*}$-parabolic manifold. Polynomials on $X$ are functions $f\in {\mathcal O}(X)$ satisfying the inequality $\log|f|\leqslant C+d\cdot \rho(z)$ for some $d\in {\mathbb N}$. We show that they can only be trivial, in the sense that $f=\mathrm{const}$. This is a consequence of the following proposition, which looks obvious, although we only prove it for $K$ constructed in Theorem 2.1: if $f(z)\in {\mathcal O}({{\{|z|<3)\}}}\setminus K)$ and
$$
\begin{equation}
\varlimsup_{z\to K}|f(z)|\cdot \operatorname{dist} (z,K)=0,
\end{equation}
\tag{2.14}
$$
then $f(z)\in {\mathcal O}(\{|z|<3)\})$. In fact, consider the closed curve $\gamma=\gamma_{m}$ formed by the straight line $\{\operatorname{Im} z=r\}$, $r>0$, on the top, the line $\{\operatorname{Im} z=-r\}$ on the bottom, and segments of the lines $\{\operatorname{Re} z=a_{mj} -r\}$ and $\{\operatorname{Re} z=b_{mj} +r\}$ on the sides. Clearly, if $r>0$ is sufficiently small, then $\gamma $ contains the compact set
$$
\begin{equation*}
K\subset K_{m}=[a_{m1}, b_{m1}]\cup [a_{m2}, b_{m2}]\cup\dots \cup [a_{m2^{m}}, b_{m2^{m}}]
\end{equation*}
\notag
$$
inside it and has length
$$
\begin{equation}
l(\gamma)=2\cdot 2^{m} (\delta^{t_{m}} +2r)+2\cdot 2^{m} r=3\cdot 2^{m+1} r+2^{m+1} \delta^{t_{m}}.
\end{equation}
\tag{2.15}
$$
Now we use Cauchy’s formula
$$
\begin{equation}
f(z)=\frac{1}{2\pi i} \int_{|\xi|=2} \frac{f(\xi)}{\xi -z} \,d\xi -\frac{1}{2\pi i} \int_{\gamma} \frac{f(\xi)}{\xi -z}\,d\xi, \qquad z\in B(0,2)\setminus \widehat{\gamma},
\end{equation}
\tag{2.16}
$$
where $\widehat{\gamma}$ is the convex polynomial hull of $\gamma $. For the second integral in (2.16) we have
$$
\begin{equation*}
\begin{aligned} \, \biggl|\int_{\gamma} \frac{f(\xi)}{\xi -z} \,d\xi \biggr| &\leqslant \frac{\| f\|_{\gamma}}{\operatorname{dist} (z,\gamma)}\cdot l(\gamma)\leqslant \frac{e^{C+d\cdot \| \rho \|_{\gamma}}}{\operatorname{dist} (z,\gamma)} (3\cdot 2^{m+1} r+2^{m+1} \delta^{t_{m}}) \\ &\leqslant C_{1} e^{d\cdot \| \rho \|_{\gamma}} \cdot (2^{m+3} r+2^{m+1} \delta^{t_{m}}). \end{aligned}
\end{equation*}
\notag
$$
By (2.1), for any fixed $\varepsilon\,{>}\,0$ there exists $\gamma\!=\!\gamma_{m}$ such that $\| \rho \|_{\gamma} \!<\!-\varepsilon \log\operatorname{dist} (\gamma, K)$. Hence
$$
\begin{equation*}
\biggl|\int_{\gamma} \frac{f(\xi)}{\xi -z}\,d\xi \biggr| \leqslant C_{2} r^{-d\varepsilon} 2^{m} (r+\delta^{t_{m}}).
\end{equation*}
\notag
$$
Now we can select $\varepsilon=1/(2d)$ and $r=1/2^{4m}$. Then
$$
\begin{equation*}
r^{-\varepsilon d} 2^{m} (r+\delta^{t_{m}})=\frac{1}{2^{m}}+2^{3m} \delta^{t_{m}} \to 0 \quad\text{as } m\to\infty.
\end{equation*}
\notag
$$
We see that the second integral in (2.16) tends to zero, which means that the function
$$
\begin{equation*}
f(z)=\frac{1}{2\pi i}\int_{|\xi |=2} \frac{f(\xi)}{\xi -z}\,d\xi
\end{equation*}
\notag
$$
is holomorphic in the disc $|z|<2$ and therefore also in the disc $|z|<3$. It now follows that if $f\in {\mathcal P}_{\rho}^{} (X)$, then $f\in {\mathcal O}(\overline{\mathbb C})$, that is, $f\equiv \mathrm{const}$. We have demonstrated that ${\mathcal P}_{\rho}^{} (X)$ consists of constants. Example 2.2. Consider an algebraic variety $A\subset{\mathbb C}^{N}$, $\dim A=n$. By Rudin’s well-known criterion of the algebraicity of an analytic set [19] we can assume that (after a suitable affine transformation) where $C$ and $d$ are some constants. Note that in [20] the first-named author showed that we can take $d=1$. Thus, if we set $\rho (w)=\log \| w'\|$, then the restriction $\rho |_{A}$ is a special exhaustion function on $A$. Clearly, $\rho$-polynomials on $A$ are the restrictions of polynomials $p(w{'}, w{''})$ on ${\mathbb C}^{N}$. Hence ${\mathcal P}_{\rho} (A)$ is dense in ${\mathcal O}(A)$. Definition 2.1. An $S$-parabolic manifold $X$ is said to be regular if the space of all $\rho$-polynomials on ${\mathcal P}_{\rho}(X)$ is dense in ${\mathcal O}(X)$. § 3. Polynomial approximation and rate of convergence on regular parabolic manifoldsIn 1962 Siciak [21] (§ 10, Theorems 1 and 2) proved the following generalization of the classical Bernstein–Walsh theorem to ${\mathbb C}^{n}$. Theorem 3.1 (Siciak). Let $K\subset {\mathbb C}^{n}$ be a regular compact set and be the least deviation of the function $f$ on $K$ from the class of polynomials ${\mathcal P}^{d} ({\mathbb C}^{n})$. Then the function $f(z)$, defined originally on $K$, extends holomorphically to a neighbourhood $D_{R}=\{z\in X\colon \Phi (z,K)<R\}$, $R>1$, if and only if Here
$$
\begin{equation*}
\Phi (z,K)=\sup \{|P(z)|^{{1}/{\deg P}}\colon P\in {\mathcal P}({\mathbb C}^{n}),\,\| f\|_{K} \leqslant 1\}
\end{equation*}
\notag
$$
is the Green’s extremal function for $K$. Similar results on rapid (geometric) approximation of analytic functions by polynomials or rational functions were also obtained by Gonchar [22], Chirka [23], Zakharyuta [24], Sadullaev [25], [26] and many other authors. In this section we discuss rapid approximation on parabolic Stein manifolds and prove an analogue of the Bernstein–Walsh theorem. Let $X$ be an $S$-parabolic manifold and $\rho (z)$ be a special exhaustion function. Let ${\mathcal P}_{\rho}^{d} (X)$ denote the class of polynomials of degree $\leqslant d$ on $X$. Let
$$
\begin{equation*}
e_{d} (f,K)=\inf_{P\in {\mathcal P}_{\rho}^{d} (X)} \sup_{z\in K} |f(z)-P(z)|
\end{equation*}
\notag
$$
be the least deviation of $f$ from the space of polynomials ${\mathcal P}_{\rho}^{d} (X)$ on the compact set $K$. In a similar way we define the extremal Green’s function on an $S$-parabolic manifold by
$$
\begin{equation*}
\Phi (z,K)=\sup \{|P(z)|^{{1}/{\deg P}}\colon P\in {\mathcal P}_{\rho}^{} (X), \,\|f\|_{K} \leqslant 1\}.
\end{equation*}
\notag
$$
It is obvious that $\Phi (z,K)$ has the following properties.
We use the following well-known concept of regularity for compact sets in ${\mathbb C}^{n}$. Definition 3.1. A compact subset $K$ of $ X$ is regular if the function $\Phi (z,K)$ is continuous on the whole of $X$. The following result is central in our paper. Theorem 3.2. Let $K\subset X$ be a regular compact subset of the regular parabolic manifold $X$. Then a function $f(z)$ defined on $K$ extends holomorphically to the domain ${R}=\{z\in X\colon \Phi (z,K)<R\}$ for $R>1$ if and only if The proof is significantly different from the proof of the analogous result in ${\mathbb C}^{n}$ (Theorem 3.1), which uses Weyl’s integral formula and Hefer’s theorem (for instance, see [27], § 3.2). We use another technique of the proof, which is based on an embedding of a polynomial polyhedron in $X$ in a complex space of large dimension. Proof of Theorem 3.2. Sufficiency. We argue as in ${\mathbb C}^{n}$: if
$$
\begin{equation*}
\lim_{d\to \infty} \| f-p_{d} \|_{K}^{1/d} \leqslant \frac{1}{R},
\end{equation*}
\notag
$$
then the series
$$
\begin{equation}
p_{1} (z)+\sum_{d=1}^{\infty}(p_{d+1} (z)-p_{d} (z)),
\end{equation}
\tag{3.2}
$$
which by the Bernstein–Walsh inequality (3.1) satisfies converges locally uniformly in the domain $D_{R}=\{z\in X\colon \Phi (z,K)<R\}$, and its sum is holomorphic in $D_{R}$.
Necessity. Fix a sufficiently small $\varepsilon >0$ such that $ \log R-\varepsilon >1$. If $f(z)$ is holomorphic in the domain $D_{R}=\{z\in X\colon \Phi (z,K)<R\}$, $R>1$, then there exists a finite set of $\rho$-polynomials $p=\{p_{1}, p_{2},\dots, p_{m}\}$, $\|p_{j}\|_{K} \leqslant 1$, such that
$$
\begin{equation*}
\sup \biggl\{\frac{1}{s_{j}} \log |p_{j}(z)|,\,j=1,2,\dots,m\biggr\}>\log R-\varepsilon, \qquad z\in \partial D_{R}.
\end{equation*}
\notag
$$
Raising the $p_{j} $ to some powers if necessary, we can assume without loss of generality that all exponents $s_{j}$ are equal to some $s$. This means that by raising each polynomial in the set $p=\{p_{1}, p_{2},\dots, p_{m}\}$ to a suitable power we obtain a set of polynomials of the same degree. Setting $P_{j}=p_{j}/(e^{-\varepsilon s} R^{s})$, we ‘embed’ the polyhedral domain
$$
\begin{equation*}
\Pi=\{ z\in X\colon |P_{j} (z)|<1,\,j=1,2,\dots,m\} \subset G_{R} \subset X\subset {\mathbb C}^{N},\qquad\dim X=n,
\end{equation*}
\notag
$$
in ${\mathbb C}^{m}$ by the map $\pi=(P_{1},P_{2},\dots,P_{m})\colon \Pi \to {\mathbb C}^{m}$. However, this map is not necessarily injective: distinct points can be glued together. In this case, assuming without loss of generality that $ D_{R} \Subset \{|w_1|< 1,\,|w_2|< 1,\dots, |w_N|< 1\}$, we add to the polynomials $\{P_{1},P_{2},\dots,P_{m} \}$ the functions $P_{m+j}={w_j|_X}/(2e^{-\varepsilon s} R^s)$, ${j=1,2,\dots,N}$, where $(w_{1}, w_{2},\dots, w_{N})$ are the coordinate variables in the space ${\mathbb C}^{N} \supset X$ and ${w_j|_{X}}$ is the restriction of $w_j$ to $X$. Then the map
$$
\begin{equation*}
\pi '=\{P_{1}, P_{2},\dots, P_{m}, P_{m+1}, P_{m+2},\dots, P_{m+N} \}\colon \Pi \to {\mathbb C}^{m+N}
\end{equation*}
\notag
$$
maps the polyhedral domain $ \Pi $ univalently to the polydisc Since the vector function $(w_1, w_2,\dots,w_N)$ separates points in $X$, the map $\pi'$: ${\Pi \to U^{m+N}}$ is a holomorphic embedding, and its image $ \pi ' (\Pi) $ is an $n$-dimensional closed submanifold of $U^{m+N}$.
Note that the coordinate variables $w_j$ are not necessarily $\rho$-polynomials on $X$. However, since $X$ is regular, the functions $P_{m+j}$ are arbitrarily close to $\rho$-polynomials on compact subsets of $X$. Hence we can replace the $P_{m+j}$ by some $\rho$-polynomials $P'_{m+j}$ while preserving the univalence of the map
$$
\begin{equation*}
\{P_{1}, P_{2},\dots, P_{m}, P'_{m+1}, P'_{m+2},\dots, P'_{m+N} \}\colon \Pi \to U^{m+N}
\end{equation*}
\notag
$$
and the inequalities
$$
\begin{equation*}
|P'_{m+j} {z}|< \frac{1}{e^{-\varepsilon s} R^s}, \qquad z \in D_R.
\end{equation*}
\notag
$$
We denote the new map $\{P_{1},P_{2},\dots,P_{m},P'_{m+1},P'_{m+2},\dots, P'_{m+N}\}$ by $\pi'=\{P_{1},P_{2},\dots, P_{m}, P_{m+1}, P_{m+2},\dots, P_{m+N} \} $ again. Raising to suitable powers again we can assume that all polynomials $ P_{j}$, $j=1,2,\dots,m+N$, have the same degree, equal to $s$ say.
By the Oka–Cartan extension theorem (see [28], Ch. 2, § 2.6, Corollary 2.6.3) each analytic function on $ \pi'(\Pi)$ extend holomorphically to the polydisc $U^{m+N} \subset {\mathbb C}^{m+N}$, that is, for each holomorphic function $f(z)\in{\mathcal O}(D_{R})$ there exists a holomorphic function $g(v)=g(v_{1}, v_{2},\dots, v_{m+N})\in {\mathcal O}(U^{m+N})$ such that $g|_{\pi' (\Pi)}=f$. Now we expand $g(v)=g(v_{1}, v_{2},\dots, v_{m+N})\in {\mathcal O}(U^{m+N})$ in a Taylor series:
$$
\begin{equation}
g(v)=\sum_{|k|=0}^{\infty}c_{k}\cdot v_{1}^{k_{1}}\cdot v_{2}^{k_{2}}\dotsb v_{m+N}^{k_{m+N}}, \qquad |k|=k_{1} +k_{2} +\dots +k_{m+N}.
\end{equation}
\tag{3.3}
$$
Fix a sufficiently small positive real number $\sigma<1$ such that
$$
\begin{equation*}
U_{1-\sigma}^{m+N}=\{|v_1|<1-\sigma,\,|v_2|<1-\sigma,\dots,|v_{m+N}|<1-\sigma\} \supset\pi ' (K).
\end{equation*}
\notag
$$
If we set $M_\sigma=\|g\|_{\overline {U}_{1-\sigma}^{m+N}}$, then the following Cauchy inequality holds for the coefficients of (3.3):
$$
\begin{equation}
|c_{k}|\leqslant \frac{M_{\sigma}}{(1-\sigma)^{|k|}}, \qquad |k|=k_{1} +k_{2} +\dots +k_{m+N}.
\end{equation}
\tag{3.4}
$$
Substituting $v_{j}=P_{j}(z)$, $j=1,2,\dots,m + N$, into (3.3) we obtain the expansion
$$
\begin{equation}
f(z)=\sum_{|k|=0}^{\infty}c_{k}\cdot P_{1}^{k_{1}}(z)\cdot P_{2}^{k_{2}} (z)\dotsb P_{m+N}^{k_{m+N}} (z), \qquad z\in \Pi .
\end{equation}
\tag{3.5}
$$
A partial sum
$$
\begin{equation*}
Q_{t} (z)=\sum_{|k|=0}^{q}c_{k}\cdot P_{1}^{k_{1}}(z)\cdot P_{2}^{k_{2}} (z)\dotsb P_{m+N}^{k_{m+N}}(z)
\end{equation*}
\notag
$$
is a $\rho$-polynomial of degree $t=sq$. From (3.2), using estimates (3.4) we obtain
$$
\begin{equation*}
\| f-Q_{t} \|_{K} \leqslant \sum_{|k|=q+1}^{\infty}\frac{M_{\sigma}}{(1-\sigma)^{|k|}}\cdot \frac{1}{(e^{-\varepsilon} R)^{s|k|}}=M_{\sigma} \sum_{|k|=q+1}^{\infty} \frac{1}{((1-\sigma)e^{-s\varepsilon} R^s)^{|k|}}.
\end{equation*}
\notag
$$
Hence for $(1-\sigma)e^{-s\varepsilon} R^s >1$ we have
$$
\begin{equation*}
\| f-Q_{t} \|_{K} \leqslant C(\sigma, \varepsilon, s){((1-\sigma) e^{-s\varepsilon} R^s)}^{-q},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
C(\sigma,\varepsilon,s)=M_{\sigma}\sum_{|k|=0}^{\infty}\frac{1}{((1-\sigma) e^{-\varepsilon s} R^s)^{|k|}}.
\end{equation*}
\notag
$$
This yields the inequality
$$
\begin{equation*}
\varlimsup_{\substack{q\to \infty\\ t=sq}} \| f-{{Q}_{t}} \|_K^{1/t}\leqslant \frac{1}{(1-\sigma)^{1/s}e^{-\varepsilon}R}.
\end{equation*}
\notag
$$
Since $\varepsilon >0$ and $\sigma>0$ are arbitrary, it follows that
$$
\begin{equation*}
\varlimsup_{q\to \infty}e_{sq}^{1/(sq)} (f,K) \leqslant \frac{1}{R}.
\end{equation*}
\notag
$$
From the subsequence $\{sq\}$ we can obviously proceed to the sequence $\{d\}$ itself, that is,
$$
\begin{equation*}
\varlimsup_{d\to \infty}e_{d}^{1/d} (f,K) \leqslant \frac{1}{R},
\end{equation*}
\notag
$$
because for $0\leqslant k \leqslant s$ we have
$$
\begin{equation*}
e_{sq+k}^{1/(sq+k)} \leqslant e_{sq}^{1/(sq+s)} \leqslant [e_{sq}^{1/(sq)}]^{sq/(sq+s)}.
\end{equation*}
\notag
$$
Theorem 3.2 is proved. Theorem 3.2 has the following corollary, analogous to Runge’s approximation theorem. Corollary 3.1. Let $X$ be a regular parabolic manifold with special exhaustion function $\rho (z)$. If $K$ is a $\rho$-polynomial convex compact subset of $X$, then each analytic function $ f(z)$ in a neighbourhood of $K$ can be approximated by $\rho$-polynomials uniformly on $K$. The main results of this paper were reported on October 31, 2022, at the international conference dedicated to the 80th birthday of professor Evgenii Mikhailovich Chirka at the Steklov Mathematical Institute of the Russian Academy of Sciences. 
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\Bibitem{SadAta24} |
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