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<title>Strong asymptotics of extremal polynomials on the Segment in the presence of denumerable set of mass points: Strong asymptotics of extremal polynomials on the Segment in the presence of denumerable set of mass points</title>
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<h1>Strong asymptotics of extremal polynomials on the Segment in the presence of denumerable set of mass points</h1>
<p class="authors">
<span class="author">Rabah Khaldi\(^\ast \) Ahcene Boucenna\(^\ast \)</span>
</p>
<p class="date">October 6, 2010.</p>
</div>
<p>\(^\ast \)Laboratory LASEA, Faculty of Sciences, University of Annaba, B.P.12, 23000, Annaba, Algeria, e-mail: <span class="ttfamily">{rkhadi@yahoo.fr, ahcene03081977@yahoo.fr}</span>. </p>

<div class="abstract"><p> The strong asymptotics of the monic extremal polynomials with respect to a \(L_{p}(\sigma )\) norm are studied. The measure \(\sigma \) is concentrated on the segment \([-1,1]\) plus a denumerable set of mass points which accumulate at the boundary points of \([-1,1]\) only. Under the assumptions that the mass points satisfy Blaschke’s condition and that the absolutely continuous part of \(\sigma \) satisfies Szegő’s condition. </p>
<p><b class="bf">MSC.</b> 42C05, 30E15, 30E10 </p>
<p><b class="bf">Keywords.</b> Orthogonal polynomials, asymptotic behavior. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>Let \(0{\lt}p{\lt}\infty \) and \(\sigma \) be a positive Borel measure supported on an infinite compact set \(E\) of the complex plane. We can then define for \(n=1,2,3,...\).</p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  m_{n,p}\left( \sigma \right) :=\underset {Q\in \mathcal{P}_{n-1}}{\min }\left\Vert z^{n}-Q\left( z\right) \right\Vert _{L_{p}\left( \sigma \right) }, \end{equation*}
</div>
<p>where \(\mathcal{P}_{n-1}\) denotes the class of complex polynomials of degree at most \(n-1\). It is easily seen that there is at least one monic polynomial <br />\(T_{n,p}(\sigma ,z)=z^{n}+...\in \mathcal{P}_{n}\) such that</p>
<div class="equation" id="1">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert T_{n,p}(\sigma ,z)\right\Vert _{L_{p}\left( \sigma \right) }=m_{n,p}\left( \sigma \right) . \label{1} \end{equation}
  </div>
  <span class="equation_label">1</span>
</p>
</div>
<p>We call \(T_{n,p}(\sigma ,z)\) an \(L_{p}\) extremal polynomial with respect to the measure \(\sigma \). We define also the nomalized extremal polynomials</p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  P_{n,p}(\sigma ,z):=T_{n,p}(\sigma ,z)/m_{n,p}\left( \sigma \right) , \end{equation*}
</div>
<p> \(n=1,2,3,...\), satisfying</p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  \left\Vert P_{n,p}(\sigma ,z)\right\Vert _{L_{p}\left( \sigma \right) }=1. \end{equation*}
</div>
<p>When \(p=2\), \(P_{n,2}(\sigma ,z)=\mathbb {\kappa }_{n}z^{n}+...\in \mathcal{P}_{n}\) \(\left( \mathbb {\kappa }_{n}=1/m_{n,2}\left( \sigma \right) {\gt}0\right) \) is just the orthonormal polynomial of degree \(n\) with respect to the measure \(\sigma \) i.e. </p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  (P_{n,2},z^{k})_{L_{p}\left( \sigma \right) }:=\int _{E}P_{n,2}(\xi )\overline{\xi }^{k}{\rm d}\sigma (\xi )=\mathbb {\kappa }_{n}^{-1}\delta _{nk},k=0,1,...,n. \end{equation*}
</div>
<p>A special area of research in this subject has been the study of the asymptotic behavior of \(T_{n,p}(z)\) when \(n\) tends to infinity. There exists an extensive literature on orthogonal polynomials, but not enough on extremal polynomials. Beginning by Geronimus results in 1952 [1], who considered the case where the support \(E\) of the measure is a rectifiable Jordan curve, in particular, Widom [11] investigated the case \(p=\infty \). Then, in 1987, Lubinsky and Saff [7] proved the asymptotic of \(m_{n,p}\left( \sigma \right) \) and \(T_{n,p}\) outside the segment \([-1,1]\) under a general condition on the weight function. Another result on the zero distributions of the extremal polynomials on the unit circle, was presented by X. Li and K. Pan in [6]. In 1992, Kaliaguine [2], obtained the power asymptotic for extremal polynomials when \(E\) is a rectifiable Jordan curve plus a finite set of mass points and in 2004, Khaldi presented in [4] an extension of Kaliaguine’s results, where he studied the case of a measure supported on a rectifiable Jordan curve plus an infinite set of mass points. Recently, Khaldi [5], solved this problem for a measure supported on the segment \([-1,1]\) plus a finite set of mass points. </p>
<p>We mentioned that in the special case \(p=2\) of orthogonal polynomials, Peherstorfer and Yudiskii in [8] established the asymptotic for such polynomial on a segment \(\left[ -2,+2\right] \) plus a infinite set of mass points. </p>
<p>In this paper, we generalize the work of Peherstorfer and Yudiskii in [8] in the case where \(p\geq 2\), more precisely we establish the strong asymptotic of the \(L_{p}\) extremal polynomials \(\left\{  T_{n,p}(\sigma ,z)\right\}  \) associated with the measure \(\sigma \) which has a decomposition of the form \(\sigma =\alpha +\gamma \), where \(\alpha \) is a measure with supp\((\alpha )=[-1,1]\), absolutely continuous with respect to the Lebesgue measure on the segment \([-1,1]\) i.e. </p>
<div class="equation" id="2">
<p>
  <div class="equation_content">
    \begin{equation}  {\rm d}\alpha (x)=\rho (x){\rm d}x,\quad \rho \geq 0,\quad \int _{-1}^{+1}\rho (x){\rm d}x<+\infty , \label{2} \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>satisfying Szegő’s condition and \(\gamma \) is a discrete measure supported on the infinite set of points \(\left\{  z_{k}\right\}  _{k=1}^{\infty }\subset \mathbb {C}\backslash \lbrack -1,+1]\) i.e. </p>
<div class="equation" id="3">
<p>
  <div class="equation_content">
    \begin{equation}  \gamma =\sum \limits _{k=1}^{\infty }A_{k}\delta (z-z_{k});\quad A_{k}>0,\quad \sum \limits _{k=1}^{\infty }A_{k}<\infty . \label{3} \end{equation}
  </div>
  <span class="equation_label">3</span>
</p>
</div>
<h1 id="a0000000007">2  Preliminary materials</h1>
<h2 id="a0000000008">2.1 Hardy space and Szegő function</h2>
<p>Let \(E=[-1,1],\) \(\Omega =\left\{  \mathbb {C}\backslash E\right\}  \cup \left\{  \infty \right\}  ,\) \(G=\{ w\in \mathbb {C}:\left\vert w\right\vert {\gt}1\} \cup \left\{  \infty \right\}  .\) The conformal mapping \(\Phi :\Omega \rightarrow G\) is defined by \(\Phi (z)=z+\sqrt{z^{2}-1},\) its inverse\(\, \Psi (w)=\tfrac {1}{2}\left( w+\tfrac {1}{w}\right) ,\) and the capacity \(C(E)=\underset {z\rightarrow \infty }{\lim }\left( \tfrac {z}{\Phi (z)}\right) =\tfrac {1}{2}.\) </p>
<p>Let \(\rho \) be an integrable non negative weight function on \(E\) satisfying the Szegő’s condition </p>
<div class="equation" id="4">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{-1}^{1}\tfrac {{\rm Log}\rho (x)}{\sqrt{1-x^{2}}}{\rm d}x>-\infty . \label{4} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>Then we can easily see that the weight function \(\lambda \) defined on the unit circle by </p>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  \lambda (e^{i\theta })=\left\{  \begin{array}{c} \rho (\xi )/\left\vert \Phi _{-}^{^{\prime }}(\xi )\right\vert ,\quad \xi =\Psi (e^{i\theta }),\pi {\lt}\theta {\lt}2\pi \\ \rho (\xi )/\left\vert \Phi _{+}^{^{\prime }}(\xi )\right\vert ,\quad \xi =\Psi (e^{i\theta }),0{\lt}\theta {\lt}\pi \end{array}\right. \end{equation*}
</div>
<p>satisfies the following usual Szegő’s condition </p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  \int _{-\pi }^{\pi }{\rm Log}(\lambda (e^{{\rm i}\theta })){\rm d}\theta {\gt}-\infty . \end{equation*}
</div>
<p>Thus the Szegő function associated with the unit circle \(T=\left\{  t:\left\vert t\right\vert =1\right\}  \) and the weight function \(\lambda \) is defined by </p>
<div class="equation" id="5">
<p>
  <div class="equation_content">
    \begin{equation}  D\left( w\right) =\exp \left\{  -\tfrac {1}{2p\pi }\int _{0}^{2\pi }{\rm Log}(\rho (\cos \theta )\left\vert \sin \theta \right\vert )\tfrac {1+we^{-{\rm i}\theta }}{1-we^{-{\rm i}\theta }}{\rm d}\theta \right\}  ,\left\vert w\right\vert <1, \label{5} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>satisfying the following properties: </p>
<p>1) \(D\) is analytic on the open unit disk \(U=\left\{  w:\left\vert w\right\vert {\lt}1\right\}  ,\) \(D\left( w\right) \neq 0,\) \(\forall w\in U\), and \(D\left( 0\right) {\gt}0.\) </p>
<p>2) \(D\) has boundary values, almost everywhere on the unit circle \(T\) such that </p>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  \lambda (e^{{\rm i}\theta })=\rho (\cos \theta )\left\vert \sin \theta \right\vert =\left\vert D(t)\right\vert ^{-p} \end{equation*}
</div>
<p> a.e. for \(t=e^{{\rm i}\theta }\in T\). </p>
<p><div class="definition_thmwrapper " id="a0000000012">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.1</span>
  </div>
  <div class="definition_thmcontent">
  <p>An analytic function \(f\) on \(\Omega \), belongs to \(H^{p}(\Omega ,\rho )\) if and only if \(f(\Psi (w))/D(1/w)\in H^{p}(G),\)where \(H^{p}(G)\) is the usual Hardy space associated with \(G\), the exterior of the unit circle. </p>

  </div>
</div> </p>
<p>Any function \(f\in H^{p}(\Omega ,\rho )\) has boundary values \(f_{+}\) and \(f_{-}\) on both sides of \(E\), and<i class="itshape"> </i>\(f_{+}\), \(f_{-}\in L_{p}(\alpha )\). </p>
<p>In the Hardy space \(H^{p}(\Omega ,\rho )\) we will define </p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  \left\Vert f\right\Vert _{H^{p}(\Omega ,\rho )}^{p}=\oint _{E}\left\vert f(x)\right\vert ^{p}\rho (x){\rm d}x=\lim \limits _{R\rightarrow 1^{+}}\tfrac {1}{\pi R}\int \nolimits _{E_{R}}\tfrac {\left\vert f\left( z\right) \right\vert ^{p}}{\left\vert D\left( z\right) \right\vert ^{p}}\left\vert \Phi ^\prime \left( z\right) \right\vert \left\vert {\rm d}z\right\vert , \end{equation*}
</div>
<p> where \(E_{R}=\left\{  z\in \Omega :\left\vert \Phi (z)\right\vert =R\right\}  . \) </p>
<h2 id="a0000000014">2.2 Notations and lemmas</h2>
<p>Let \(1\leq p{\lt}\infty \). We denote by \(\mu \left( \rho \right) \) and\(\  \mu \left( \sigma \right) \) respectively the extremal values of the following problems: </p>
<div class="equation" id="6">
<p>
  <div class="equation_content">
    \begin{equation}  \mu \left( \rho \right) =\inf \left\{  \left\Vert \varphi \right\Vert _{H^{p}\left( \Omega ,\rho \right) }^{p}:\varphi \in H^{p}\left( \Omega ,\rho \right) ,\varphi \left( \infty \right) =1\right\}  , \label{6} \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<div class="equation" id="7">
<p>
  <div class="equation_content">
    \begin{equation}  \mu \left( \sigma \right) =\inf \left\{  \left\Vert \varphi \right\Vert _{H^{p}\left( \Omega ,\rho \right) }^{p}:\varphi \in H^{p}\left( \Omega ,\rho \right) ,\varphi \left( \infty \right) =1,\varphi \left( z_{k}\right) =0,k=1,2,...\right\}  \label{7} \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>We denote by \(\varphi ^{\ast }\) and \(\psi ^{\ast }\) the extremal functions of the problems (6) and (7) respectively. </p>
<p>Notice that \(\varphi ^{\ast }(z)=D\left( 1/\Phi (z)\right) /D\left( 0\right) \) is an extremal function of the problem (6) and \(\mu (\rho )=2/\left[ D\left( 0\right) \right] ^{p}\) (see [3]). </p>
<p><div class="lem_thmwrapper " id="a0000000015">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">2.2</span>
  </div>
  <div class="lem_thmcontent">
  <p>The extremal functions \(\varphi ^{\ast }\) and \(\psi ^{\ast }\) are related by </p>
<p>\(\psi ^{\ast }=\tfrac {1}{B(\infty )}B\varphi ^{\ast }\) and \(\mu (\sigma )=\left[ B(\infty )\right] ^{-p}\mu (\rho ),\) where </p>
<div class="equation" id="8">
<p>
  <div class="equation_content">
    \begin{equation}  B(z)=\prod \limits _{k=1}^{\infty }\tfrac {\Phi (z)-\Phi (z_{k})}{\Phi (z)\overline{\Phi (z_{k})}-1}\tfrac {\left\vert \Phi (z_{k})\right\vert }{\Phi (z_{k})} \label{8} \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>is a Blaschke product. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000016">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>This lemma is proved for a curve in [4, p. 374]. This proof is valid in this case, too. </p>
<p><div class="definition_thmwrapper " id="a0000000017">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.3</span>
  </div>
  <div class="definition_thmcontent">
  <p>A measure \(\sigma =\alpha +\sum _{k=1}^{\infty }A_{k}\delta (z-z_{k})\) is said to belong to a class \(A\), if the absolutely continuous part \(\alpha \) satisfies the Szegö’s condition <span class="rm">(4)</span> and the discrete part satisfies the the Blaschke’s condition</p>
<div class="equation" id="9">
<p>
  <div class="equation_content">
    \begin{equation}  \left( \underset {k=1}{\overset {\infty }{\sum }}\left\vert \Phi (z_{k})\right\vert -1\right) <\infty . \label{9} \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="lem_thmwrapper " id="a0000000018">
  <div class="lem_thmheading">
    <span class="lem_thmcaption">
    Lemma
    </span>
    <span class="lem_thmlabel">2.4</span>
  </div>
  <div class="lem_thmcontent">
  <p>Let \(\sigma =\alpha +\sum _{k=1}^{\infty }A_{k}\delta (z-z_{k})\) be a measure which belongs to a class \(A\), then we have</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \underset {n\rightarrow \infty }{\limsup }\;  2^{n}m_{n,p}(\sigma )\leq \left[ \mu \left( \sigma \right) \right] ^{1/p}. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000020">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>This lemma is proved for \(p=2\) by Peherstorfer and Yudiskii in [8]. We will prove this lemma for \(p{\gt}2\) following the same ideas as in [8]. </p>
<p>Without loss of generality, we assume the boundness from below of the weight function \(\tfrac {1}{\left\vert D\right\vert },\) so \(\tfrac {1}{\left\vert D\right\vert }\geq 2\). Let \(\tfrac {1}{\left\vert D_{\varepsilon }\right\vert } \) be a smooth function such that \(\tfrac {1}{\left\vert D_{\varepsilon }\right\vert }\geq 1\) and </p>
<div class="equation" id="10">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{T}\left\vert \tfrac {1}{\left\vert D_{\varepsilon }\right\vert ^{p}}-\tfrac {1}{\left\vert D_{\varepsilon }\right\vert ^{p}}\right\vert {\rm d}m<\epsilon \label{10} \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>for \(\varepsilon {\gt}0.\) Let us choose \(\eta {\gt}0\) such that \(\max \tfrac {1}{\left\vert D_{\varepsilon }\right\vert }\leq \tfrac {1}{\eta }\) and denote by \(E_{\pm }\) and\(\  \widetilde{E}_{\pm }\) the vicinities of \(\pm 1\) of the form </p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  E_{\pm }=\left\{  t\in T,\left\vert t\pm 1\right\vert \leq \tfrac {\eta }{2}\right\}  ,\quad \widetilde{E}_{\pm }=\left\{  t\in T,\left\vert t\pm 1\right\vert \leq \eta \right\}  . \end{equation*}
</div>
<p>Introduce a smooth function as follows</p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  F_{\varepsilon ,\eta }\left( t\right) =\left\{  \begin{array}{ll} \left\vert D_{\varepsilon }\left( t\right) \right\vert ,&  t\in T\backslash \widetilde{E}_{+}\cup \widetilde{E}_{-} \\ \left\vert t\pm 1\right\vert ^{2}, &  t\in E_{\pm }\end{array}\right. \end{equation*}
</div>
<p>and for \(t\in \) \(\widetilde{E}_{\pm }\backslash E_{\pm }\) is such that </p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  \left\vert t\pm 1\right\vert ^{2}\leq \left\vert F_{\varepsilon ,\quad \eta }\left( t\right) \right\vert \leq \left\vert D_{\varepsilon }\left( t\right) \right\vert . \end{equation*}
</div>
<p>By the above settings it yields</p>
<div class="displaymath" id="a0000000024">
  \begin{eqnarray*}  0 & \leq & \log \tfrac {1}{F_{\varepsilon ,\eta }\left( 0\right) }-\log \tfrac {1}{D_{\varepsilon }(0)}\leq \int _{\widetilde{E}_{+}\cup \widetilde{E}_{-}}\log \left\vert \tfrac {D_{\varepsilon }\left( t\right) }{F_{\varepsilon ,\eta }\left( t\right) }\right\vert {\rm d}m \\ & \leq & \int _{\widetilde{E}_{+}}\log \tfrac {1}{\left\vert t+1\right\vert ^{2}}{\rm d}m+\int _{\widetilde{E}_{-}}\log \tfrac {1}{\left\vert t-1\right\vert ^{2}}{\rm d}m=o(1),\quad {\rm as }\  \eta \rightarrow 0 \end{eqnarray*}
</div>
<p>In view of (10), we get</p>
<div class="equation" id="11">
<p>
  <div class="equation_content">
    \begin{equation}  F_{\varepsilon ,\eta }\left( 0\right) =D\left( 0\right) +o\left( 1\right) ;\quad \varepsilon \rightarrow 0,\eta \rightarrow 0. \label{11} \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>Let \(b(t)\) be the Blaschke product</p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  b(t)=\prod \limits _{k=1}^{\infty }\tfrac {t-t_{k}}{t\overline{t_{k}}-1}\tfrac {\overline{t_{k}}}{\left\vert t_{k}\right\vert } \end{equation*}
</div>
<p>with \(t_{k}=\tfrac {1}{\Phi (z_{k})},k=0,1,2,...\). We see that \(b(t)\) oscillates only in vicinities of the points \(\pm 1\), moreover, we have</p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  \sup \left\{  \left\vert b^{\prime }(t)\left\vert t^{2}-1\right\vert ^{2},t\in T\right\vert \right\}  {\lt}\infty , \end{equation*}
</div>
<p>Consequently, \(\left( bF_{\varepsilon ,\eta }\right) ^{\prime }=b^{\prime }F_{\varepsilon ,\eta }+bF_{\varepsilon ,\eta }^{\prime }\in L_{\infty }\), and the Fourier series of \(bF_{\varepsilon ,\eta }\) converges to this function uniformly on \(T\). Let</p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \left( bF_{\varepsilon ,\eta }\right) \left( t\right) =Q_{n,\varepsilon ,\eta }\left( t\right) +t^{n+1}g_{n,\varepsilon ,\eta }\left( t\right) ,g_{n,\varepsilon ,\eta }\in H^{\infty }. \end{equation*}
</div>
<p>Putting </p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  2^{n}P_{n,\varepsilon ,\eta }\left( z\right) =2^{n}P_{n,\varepsilon ,\eta }\left( \Psi \left( \xi \right) \right) =\tfrac {\zeta ^{-n}Q_{n,\varepsilon ,\eta }\left( \zeta \right) +\zeta ^{n}Q_{n,\varepsilon ,\eta }\left( \tfrac {1}{\zeta }\right) }{2^{1/p}}. \end{equation*}
</div>
<p>We have the following estimate of the norm of the polynomial \(2^{n}P_{n,\varepsilon ,\eta }\left( z\right) \) for the absolutely continuous part of the measure, </p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  \left\Vert \tfrac {2^{n}P_{n,\varepsilon ,\eta }\left( z\left( t\right) \right) }{D_{\varepsilon }(t)}\right\Vert _{L_{p}}\leq \\ \left\Vert \tfrac {t^{-n}F_{\varepsilon ,\eta }\left( t\right) b(t)+t^{n}F_{\varepsilon ,\eta }\left( \overline{t}\right) b(\overline{t})}{D_{\varepsilon }(t)2^{1/p}}\right\Vert _{L_{p}}+\left\Vert \tfrac {tg_{n,\varepsilon ,\eta }\left( t\right) +\overline{t}g_{n,\varepsilon ,\eta }\left( \overline{t}\right) }{D_{\varepsilon }(t)2^{1/p}}\right\Vert _{L_{p}} \end{equation*}
</div>
<p>From the fact that \(\left\Vert g_{n,\varepsilon ,\eta }\right\Vert _{L_{\infty }}\rightarrow 0\) as \(n\rightarrow \infty \) and \(\left\vert \tfrac {F_{\varepsilon ,\eta }\left( z\left( t\right) \right) }{D_{\varepsilon }(t)}\right\vert \leq 1\), we conclude</p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  \left\Vert \tfrac {2^{n}P_{n,\varepsilon ,\eta }\left( z\left( t\right) \right) }{D_{\varepsilon }(t)}\right\Vert _{L_{p}}\leq 1+o(1). \end{equation*}
</div>
<p>Since \(P_{n,\varepsilon ,\eta }\) is uniformly bounded, using (10), we get</p>
<div class="equation" id="12">
<p>
  <div class="equation_content">
    \begin{equation}  \left\Vert \tfrac {2^{n}P_{n,\varepsilon ,\eta }\left( z\left( t\right) \right) }{D(t)}\right\Vert _{L_{p}}\leq 1+C\varepsilon +o(1). \label{12} \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>Finally by using the extremal property of the polynomials \(T_{n,p}\) and the fact that \(P_{n,\varepsilon ,\eta }\left( z\right) =\tfrac {\left( bF_{\varepsilon ,\eta }\right) \left( 0\right) }{2^{1/p}}z^{n}+...\) we get with the help of (12) </p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  2^{n}m_{n,p}(\sigma )\leq \tfrac {\left\Vert 2^{n}P_{n,\varepsilon ,\eta }\right\Vert _{L_{p}\left( \sigma \right) }}{\left( bF_{\varepsilon ,\eta }\right) \left( 0\right) /2^{1/p}}\leq \tfrac {1+C\varepsilon +o(1)}{\left( bF_{\varepsilon ,\eta }\right) \left( 0\right) /2^{1/p}}, \end{equation*}
</div>
<p>From here, lemma 1 and (11), it yields</p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  \underset {n\rightarrow \infty }{\lim \sup }2^{n}m_{n,p}(\sigma )\leq \tfrac {2^{1/p}}{b(0)D(0)}=\tfrac {\left[ \mu \left( \rho \right) \right] ^{1/p}}{B\left( \infty \right) }=\left[ \mu \left( \sigma \right) \right] ^{1/p}. \end{equation*}
</div>
<p>The proof is complete. </p>
<p>Now we give the main result of this paper: </p>
<p><div class="thm_thmwrapper " id="a0000000033">
  <div class="thm_thmheading">
    <span class="thm_thmcaption">
    Theorem
    </span>
    <span class="thm_thmlabel">2.5</span>
  </div>
  <div class="thm_thmcontent">
  <p>Let a measure \(\sigma =\alpha +\sum _{k=1}^{\infty }A_{k}\delta (z-z_{k})\) be a measure which belongs to a class \(A\). Associate with the measure \(\sigma \) the functions \(D\), \(B\) and the extremal values \(m_{n,p}(\sigma )\) and \(\mu (\sigma )\) given respectively by <span class="rm">(5), (8), (1)</span> and <span class="rm">(7)</span>. Then the monic extremal polynomials \(T_{n,p}(\sigma ,z)\) have the following asymptotic behavior as \(n\rightarrow \infty \) </p>
<ul class="itemize">
  <li><p>\(\lim 2^{n}m_{n,p}(\sigma )=\left[ \mu (\sigma )\right] ^{1/p}.\) </p>
</li>
  <li><p>\(T_{n,p}(\sigma ,z)=\left\{  \Phi (z)/2\right\}  ^{n}B(z)\tfrac {D\left( 1/\Phi (z)\right) }{D\left( 0\right) }\left[ 1+\chi _{n}(z)\right] ,\, \) where \(\chi _{n}(z)\rightarrow 0\) uniformly on compact subsets of \(\Omega \). </p>
</li>
</ul>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000034">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div></p>
<p>We recall that by putting \(t_{k}=\tfrac {1}{\Phi (z_{k})}\), \(\xi =\Psi (t)\) where \(t=e^{i\theta }\), we get \(B(\xi )=b(\overline{t})\) and \(B(\infty )=b(0)=\prod \limits _{k=1}^{\infty }\left\vert \tfrac {1}{\Phi (z_{k})}\right\vert \), with \(b(t)=\prod \limits _{k=1}^{\infty }\tfrac {t-t_{k}}{t\overline{t_{k}}-1}\tfrac {\overline{t_{k}}}{\left\vert t_{k}\right\vert }\). </p>
<p>Now we consider the following integral </p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  I_{n}=\int _{0}^{2\pi }\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}-\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) \right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }, \end{equation*}
</div>
<p> and transform it in a standard way as the following sum </p>
<div class="displaymath" id="a0000000036">
  \begin{eqnarray}  I_{n} & =& \int _{0}^{2\pi }\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }+\int _{0}^{2\pi }\left\vert b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi } \notag \\ & & -2\mathcal{R}e\int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) }\tfrac {{\rm d}\theta }{2\pi }. \label{13} \end{eqnarray}
</div>
<p>Then, applying the Hölder inequality to the first term of (13) for \(p\geq 2\) we get </p>
<div class="displaymath" id="a0000000037">
  \begin{align}  \int _{0}^{2\pi }\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi } & \leq \left( \int _{0}^{2\pi }\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\right\vert ^{p}\tfrac {{\rm d}\theta }{2\pi }\right) ^{2/p}\left( \int _{0}^{2\pi }\tfrac {{\rm d}\theta }{2\pi }\right) ^{1-2/p}\nonumber \\ & =\left[ \left( \int _{0}^{2\pi }\left\vert \tfrac {T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )}\right\vert ^{p}\left\vert D(t)\right\vert ^{-p}\tfrac {{\rm d}\theta }{\pi }\right) ^{1/p}\right] ^{2} \nonumber \\ & \leq \left[ \tfrac {1}{m_{n,p}(\sigma )}\left( \tfrac {2}{\pi }\int _{-1}^{+1}\left\vert T_{n,p}(x)\right\vert ^{p}\rho (x){\rm d}x\right) ^{1/p}\right] ^{2}\leq 2 \end{align}
</div>
<p>For the second term of (13) we transform it as the following sum</p>
<div class="displaymath" id="a0000000038">
  \begin{align*} & \int _{0}^{2\pi }\left\vert b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }=\\ & =\int _{0}^{2\pi }\left\vert b(t)\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }+ \int _{0}^{2\pi }\left\vert b(\overline{t})\right\vert ^{2}\tfrac {d\theta }{2\pi }+2\mathcal{R}e\int _{0}^{2\pi }t^{-2n}b(t)\overline{\left( \tfrac {b(\overline{t})D(\overline{t})}{D(t)}\right) }\tfrac {{\rm d}\theta }{2\pi } \\ & =2+2\mathcal{R}e\int _{0}^{2\pi }t^{-2n}b(t)\overline{\left( \tfrac {b(\overline{t})D(\overline{t})}{D(t)}\right) }\tfrac {{\rm d}\theta }{2\pi }. \end{align*}
</div>
<p>Since the last term approaches \(0\) when \(n\) tends to \(\infty \) then we have </p>
<div class="equation" id="15">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{0}^{2\pi }\left\vert \tfrac {b(t)}{D(0)}+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(0)D(t)}\right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }=2+\alpha _{n} \label{15} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p> where \(\alpha _{n}\rightarrow 0\), as \(n\rightarrow \infty \). </p>
<p>In order to estimate the last integral of (13), we transform it as follows </p>
<div class="displaymath" id="a0000000039">
  \begin{eqnarray}  J_{n} & =& \int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) }\tfrac {{\rm d}\theta }{2\pi }= \notag \\ & =& \int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{b(t)}\tfrac {d\theta }{2\pi }+ \int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{\left( \tfrac {t^{n}b(\overline{t})D(\overline{t})}{D(t)}\right) }\tfrac {{\rm d}\theta }{2\pi } \notag \\ & =& 2\int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{b(t)}\tfrac {{\rm d}t}{2\pi {\rm i}t} \notag \\ & =& 2\int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\left( \overline{b(t)}-\overline{b_{l}(t)}+\overline{b_{l}(t)}\right) \tfrac {{\rm d}t}{2\pi {\rm i}t} \notag \\ & =& 2\int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\left( \overline{b(t)}-\overline{b_{l}(t)}\right) \tfrac {{\rm d}t}{2\pi {\rm i}t}+2\int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{b_{l}(t)}\tfrac {{\rm d}t}{2\pi {\rm i}t} \label{16} \end{eqnarray}
</div>
<p>where \(b_{l}(t)=\prod \limits _{k=1}^{l}\tfrac {t-t_{k}}{t\overline{t_{k}}-1}\tfrac {\overline{t_{k}}}{\left\vert t_{k}\right\vert }\) be the finite Blaschke product with zeros \(t_{k}=\tfrac {1}{\Phi (z_{k})}\), \(k=1,2,...l\). </p>
<p>By applying the Hölder inequality to the first term of (16) we get</p>
<div class="displaymath" id="a0000000040">
  \begin{eqnarray*} & & \left\vert \int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\left( \overline{b(t)}-\overline{b_{l}(t)}\right) \tfrac {{\rm d}t}{2\pi it}\right\vert \leq \notag \\ & \leq & \left( \int _{0}^{2\pi }\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\right\vert ^{p}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{p}\left( \int _{0}^{2\pi }\left\vert b(t)-b_{l}(t)\right\vert ^{q}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{q} \notag \\ & =& \tfrac {2^{1/p}}{m_{n,p}(\sigma )}\left( \int _{0}^{2\pi }\left\vert T_{n,p}(\Psi (t))\right\vert ^{p}\left\vert D(t)\right\vert ^{-p}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{p}\left( \int _{0}^{2\pi }\left\vert b(t)-b_{l}(t)\right\vert ^{q}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{q} \notag \\ & =& \tfrac {2^{1/p+1}}{m_{n,p}(\sigma )}\left( \int _{-1}^{+1}\left\vert T_{n,p}(x)\right\vert ^{p}\rho (x){\rm d}x\right) ^\frac {1}{p}\left( \int _{0}^{2\pi }\left\vert b(t)-b_{l}(t)\right\vert ^{q}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{q} \notag \leq \end{eqnarray*}
</div>
<div class="displaymath" id="17">
  \begin{eqnarray} & \leq & 2^{1/p+1}\left( \int _{0}^{2\pi }\left\vert b(t)-b_{l}(t)\right\vert ^{q}\tfrac {{\rm d}\theta }{2\pi }\right) ^\frac {1}{q}. \label{17} \end{eqnarray}
</div>
<p>For the last term of (16) by using the residue Theorem we get </p>
<div class="displaymath" id="a0000000041">
  \begin{eqnarray} & & \int _{T}\tfrac {t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{b_{l}(t)}\tfrac {{\rm d}t}{2\pi {\rm i}t}=\int _{T}\tfrac {t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)b_{l}(t)}\left\vert b_{l}(t)\right\vert ^{2}\tfrac {{\rm d}t}{2\pi {\rm i}t} \notag \\ & =& \tfrac {2^{1/p}}{2^{n}m_{n,p}(\sigma )D(0)b_{l}(0)}+\tfrac {2^{1/p}}{m_{n,p}(\sigma )}\sum \limits _{k=1}^{l}\tfrac {t_{k}^{n-1}T_{n,p}(z_{k})}{D(t_{k})b_{l}^{^{\prime }}(t_{k})}, \label{18} \end{eqnarray}
</div>
<p>the last term of (18) can be estimated as </p>
<div class="displaymath" id="a0000000042">
  \begin{align*} & \tfrac {1}{m_{n,p}(\sigma )}\left\vert \sum \limits _{k=1}^{l}\tfrac {t_{k}^{n-1}T_{n,p}(z_{k})}{D(t_{k})b_{l}^{^{\prime }}(t_{k})}\right\vert \leq \\ & \leq \tfrac {1}{m_{n,p}(\sigma )}\left[ \sum \limits _{k=1}^{l}\left\vert T_{n,p}(z_{k})\right\vert ^{p}A_{k}\right] ^{1/p}\left[ \sum \limits _{k=1}^{l}\left( \left\vert \tfrac {1}{D(t_{k})b_{l}^{^{\prime }}(t_{k})}\right\vert \tfrac {\left\vert t_{k}^{n-1}\right\vert }{A_{k}^{1/p}}\right) ^{q}\right] ^{1/q} \\ & \leq \left[ \sum \limits _{k=1}^{l}\left( \left\vert \tfrac {1}{D(t_{k})b_{l}^{^{\prime }}(t_{k})}\right\vert \tfrac {\left\vert t_{k}^{n-1}\right\vert }{A_{k}^{1/p}}\right) ^{q}\right] ^{1/q},\  \tfrac {1}{p}+\tfrac {1}{q}=1. \end{align*}
</div>
<p>So, (18) becomes</p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \int _{0}^{2\pi }\tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\overline{b_{l}(t)}\tfrac {{\rm d}t}{2\pi {\rm i}t}=\tfrac {2^{1/p}}{2^{n}m_{n,p}(\sigma )D(0)b_{l}(0)}+\beta _{n} \end{equation*}
</div>
<p> where \(\beta _{n}\rightarrow 0\), as \(n\rightarrow \infty \) </p>
<p>So, first choosing \(l\) big enough and then \(n\) we conclude that</p>
<div class="equation" id="19">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{0}^{2\pi }\tfrac {t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}\tfrac {{\rm d}\theta }{2\pi }=\tfrac {2^{1/p}}{2^{n}m_{n,p}(\sigma )D(0)b(0)}+o(1) \label{19} \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>Substituting (14),(15) and (19) we obtain </p>
<div class="equation" id="20">
<p>
  <div class="equation_content">
    \begin{equation}  0\leq I_{n}\leq 2+2+\alpha _{n}-\tfrac {4\left( 2^{1/p}\right) }{2^{n}m_{n,p}(\sigma )D(0)b(0)}+o(1) \label{20} \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>where \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty .\) </p>
<p>Finally using the previous estimate we get </p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  \underset {n\rightarrow \infty }{\lim \inf }2^{n}m_{n,p}(\sigma )\geq \tfrac {2^{1/p}}{D(0)b(0)}=\tfrac {\left[ \mu \left( \rho \right) \right] ^{1/p}}{B\left( \infty \right) }=\left[ \mu \left( \sigma \right) \right] ^{1/p}. \end{equation*}
</div>
<p>This with Lemma 2 prove the first statement of Theorem. </p>
<p>Now, to prove (2) of Theorem, first we estimate the following integral</p>
<div class="displaymath" id="a0000000045">
  \begin{align} & \left\vert \int _{T}\left[ \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}-\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) \right] \tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}\right\vert ^{2}\leq \nonumber \\ & \leq \tfrac {1}{1-\left\vert w\right\vert }\int _{T}\left\vert \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}-\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) \right\vert ^{2}\tfrac {{\rm d}\theta }{2\pi }=\tfrac {1}{1-\left\vert w\right\vert }I_{n} \label{21} \end{align}
</div>
<p>As an immediate consequence of (20) and the first statement of Theorem, we get </p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  \underset {n\rightarrow \infty }{\lim }I_{n}=0. \end{equation*}
</div>
<p>So, from (21) yields</p>
<div class="equation" id="22">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{T}\left[ \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}-\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) \right] \tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}=o(1). \label{22} \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>On the other hand we have </p>
<div class="displaymath" id="a0000000047">
  \begin{align} & \int _{T}\left[ \tfrac {2^{1/p}t^{n}T_{n,p}(\Psi (t))}{m_{n,p}(\sigma )D(t)}-\left( b(t)+\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\right) \right] \tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}=\nonumber \\ & =\int _{T}\chi _{n}(\Psi (t))\tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}-\int _{T}\tfrac {t^{2n}b(\overline{t})D(\overline{t})}{D(t)}\tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}. \label{23} \end{align}
</div>
<p>Applying the Cauchy formula to the first term in (23), we can see that </p>
<div class="equation" id="24">
<p>
  <div class="equation_content">
    \begin{equation}  \int _{T}\chi _{n}(\Psi (t))\tfrac {1}{1-w\overline{t}}\tfrac {{\rm d}t}{2\pi {\rm i}t}=\chi _{n}(z),z=\Psi (w)\in \Omega . \label{24} \end{equation}
  </div>
  <span class="equation_label">20</span>
</p>
</div>
<p>Since the last term in (23) approaches \(0\) as \(n\rightarrow \infty \), we conclude from (22), (23) and (24), the second statement of Theorem. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
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</dd>
  <dt><a name="2">2</a></dt>
  <dd><p><i class="sc">Kaliaguine, V.A.</i>, <i class="itshape">On Asymptotics of </i>\(L_{p}\) <i class="itshape">extremal polynomials on a complex curve </i>\(\left( 0{\lt}p{\lt}\infty \right) \), J. Approx. Theory, <b class="bfseries">74</b>, pp.&#160;226–236, 1993. </p>
</dd>
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  <dd><p><i class="sc">Kaliaguine, V.A.</i>, <i class="itshape">A note on the asymptotics of orthogonal polynomials on a complex arc: the case of a measure with a discrete part</i>, J. Approx. Theory, <b class="bfseries">80</b>, pp.&#160;138–145 1995. </p>
</dd>
  <dt><a name="4">4</a></dt>
  <dd><p><i class="sc">Khaldi, R.</i>, <i class="itshape">Strong asymptotics for </i>\(L_{p}\)<i class="itshape"> extremal polynomials off a complex curve</i>, J. Appl. Math., <b class="bfseries">2004</b>, no 5, pp.&#160;371–378, 2004. </p>
</dd>
  <dt><a name="5">5</a></dt>
  <dd><p><i class="sc">Khaldi, R.</i>, <i class="itshape">Szegő asymptotics of extremal polynomials on the segment </i>\([-1,+1]\)<i class="itshape">: the case of a measure with finite discrete part</i>, Georgian Math. J., <b class="bf">14</b>, no. 4, pp.&#160;673–680, 2007. </p>
</dd>
  <dt><a name="6">6</a></dt>
  <dd><p><i class="sc">Li, X.</i> and <i class="sc">Pan, K.</i>,<i class="itshape"> Asymptotics of </i>\(L_{p}\) <i class="itshape">extremal polynomials on the unit circle</i>, J.Approx. Theory, <b class="bfseries">67</b>, pp.&#160;270–283, 1991. </p>
</dd>
  <dt><a name="7">7</a></dt>
  <dd><p><i class="sc">Lubinsky, D.S.</i> and <i class="sc">Saff, E.B.</i>, <i class="itshape">Strong asymptotics for </i>\(L_{p}\) <i class="itshape">extremal polynomials (</i>\(1{\lt}p\leq \infty \)<i class="itshape">) associated with weights on </i>\(\left[ -1,1\right] \), in Lecture Notes in Math., <b class="bfseries">1287</b>, pp.&#160;83–104, 1987. </p>
</dd>
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  <dd><p><i class="sc">Peherstorfer, F.</i> and <i class="sc">Yudiskii, P.</i>, <i class="itshape">Asymptotics of orthogonal polynomials in the presence of a denumerable set of mass points,</i> Proc. Amer. Math. Soc., <b class="bfseries">129</b>, pp.&#160;3213–3220, 2001. </p>
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</dd>
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  <dd><p><i class="sc">Szegö, G.</i>, <i class="itshape">Orthogonal plynomials</i>, 4th ed. Amer. Math. Soc. Colloquium Publ., <b class="bfseries">23</b>, Amer. Math. Soc. Providence, RI, 1975. </p>
</dd>
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</dd>
</dl>


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