\(^\ast \)Laboratory LASEA, Faculty of Sciences, University of Annaba, B.P.12, 23000, Annaba, Algeria, e-mail: {rkhadi@yahoo.fr, ahcene03081977@yahoo.fr}.

1 Introduction

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 \(n=1,2,3,...\).

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
\(T_{n,p}(\sigma ,z)=z^{n}+...\in \mathcal{P}_{n}\) such that

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

\(n=1,2,3,...\), satisfying

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.

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.

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.

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.

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.

2 Preliminary materials

2.1 Hardy space and Szegő function

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 \(C(E)=\underset {z\rightarrow \infty }{\lim }\left( \tfrac {z}{\Phi (z)}\right) =\tfrac {1}{2}.\)

Let \(\rho \) be an integrable non negative weight function on \(E\) satisfying the Szegő’s condition

Then we can easily see that the weight function \(\lambda \) defined on the unit circle by

satisfies the following usual Szegő’s condition

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

satisfying the following properties:

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 \(D\left( 0\right) {\gt}0.\)

2) \(D\) has boundary values, almost everywhere on the unit circle \(T\) such that

a.e. for \(t=e^{{\rm i}\theta }\in T\).

Any function \(f\in H^{p}(\Omega ,\rho )\) has boundary values \(f_{+}\) and \(f_{-}\) on both sides of \(E\), and \(f_{+}\), \(f_{-}\in L_{p}(\alpha )\).

In the Hardy space \(H^{p}(\Omega ,\rho )\) we will define

where \(E_{R}=\left\{ z\in \Omega :\left\vert \Phi (z)\right\vert =R\right\} . \)

2.2 Notations and lemmas

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:

We denote by \(\varphi ^{\ast }\) and \(\psi ^{\ast }\) the extremal functions of the problems (6) and (7) respectively.

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]).

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

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

Introduce a smooth function as follows

and for \(t\in \) \(\widetilde{E}_{\pm }\backslash E_{\pm }\) is such that

By the above settings it yields

In view of (10), we get

Let \(b(t)\) be the Blaschke product

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

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

Putting

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,

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

Since \(P_{n,\varepsilon ,\eta }\) is uniformly bounded, using (10), we get

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)

From here, lemma 1 and (11), it yields

The proof is complete.

Now we give the main result of this paper:

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 \(b(t)=\prod \limits _{k=1}^{\infty }\tfrac {t-t_{k}}{t\overline{t_{k}}-1}\tfrac {\overline{t_{k}}}{\left\vert t_{k}\right\vert }\).

Now we consider the following integral

and transform it in a standard way as the following sum

Then, applying the Hölder inequality to the first term of (13) for \(p\geq 2\) we get

For the second term of (13) we transform it as the following sum

Since the last term approaches \(0\) when \(n\) tends to \(\infty \) then we have

where \(\alpha _{n}\rightarrow 0\), as \(n\rightarrow \infty \).

In order to estimate the last integral of (13), we transform it as follows

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 \(t_{k}=\tfrac {1}{\Phi (z_{k})}\), \(k=1,2,...l\).

By applying the Hölder inequality to the first term of (16) we get

For the last term of (16) by using the residue Theorem we get

the last term of (18) can be estimated as

So, (18) becomes

where \(\beta _{n}\rightarrow 0\), as \(n\rightarrow \infty \)

So, first choosing \(l\) big enough and then \(n\) we conclude that

Substituting (14),(15) and (19) we obtain

where \(\alpha _{n}\rightarrow 0\) as \(n\rightarrow \infty .\)

Finally using the previous estimate we get

This with Lemma 2 prove the first statement of Theorem.

Now, to prove (2) of Theorem, first we estimate the following integral

As an immediate consequence of (20) and the first statement of Theorem, we get

So, from (21) yields

On the other hand we have

Applying the Cauchy formula to the first term in (23), we can see that

Since the last term in (23) approaches \(0\) as \(n\rightarrow \infty \), we conclude from (22), (23) and (24), the second statement of Theorem.